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GMAT Prep > Divisibility and Primes > Flashcards

Divisibility and Primes Flashcards

1
Q

Divisibility of Primes:

A

A prime number x is only divisible by x (so itself) and 1. So, the prime number x has no factors other than x and 1.

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2
Q

Divisibility and Addition/Subtraction

A

If you add or subtract multiples of an integer, you get another multiple of that integer.

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3
Q

Divisibility and Addition/Subtraction:

A
  1. If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N.

E.g.
18 + 10 = 28
(Multiple of 3) + (Non-multiple of 3) = (Non-multiple of 3)

  1. If you subtract a multiple of N from a non-multiple of N, the result is a non-multiple of N.

E.g.
18 - 10 = 8
(Multiple of 3) - (Non-multiple of 3) = (Non-multiple of 3)

  1. If you add two non-multiples of N, the result could be either a multiple of N or a non-multiple of N.

E.g.
19 + 13 = 32
(Non-multiple of 3) + (Non-multiple of 3) = (Non-multiple of 3)

19 + 14 = 33
(Non-multiple of 3) + (Non-multiple of 3) = (Multiple of 3)

So if two numbers share a factor, their sum or difference must share the same factor. This also means that in a subtraction if the two numbers don’t share any prime numbers, then their result might or might not be a prime number because the numbers don’t share any factors. But in a subtraction where both are not prime and share a prime number, their result will definitely be divisibly by that prime number as well and not be a prime.

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4
Q

Greatest Common Factor (GCF):

A

The greatest common factor is the largest divisor of two or more integers. The factor will be smaller than or equal to the starting integers.

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5
Q

Least Common Multiple (LCM):

A

The least common multiple is the smallest multiple of two or more integers. The multiple will be larger or equal to the starting integers.

when you add fractions you have to find the least common multiple.

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6
Q

Finding Greatest Common Factor and Least Common Multiple:

A
  1. Use a Venn diagram:

a. Do prime factorization.
E.g. 30 = 2 x 3 x 5
24 = 2 x 3 x 2 x 2

b. Place each shared factor into the shared area in the middle of the two-circle Venn diagram. In the above example 2 and 3 could go into the shared area. All remaining non-shared factors go into the two respective outer spaces, including the repetitions of 2s.

RESULT:
GCF is product of the primes in shared area (2x3 = 6)
LCM is product of all primes in Venn diagram (5x2x3x2x2 = 120)

REMEMBER: If two numbers have no primes in common, their GCF is 1 and their LCM is their product.

E.g. 35 (5x7) and 6 (3x2). GCF is 1 (common factor for all positive integers) and LCM is 35x6 = 210.

  1. For larger numbers use prime columns:

a. Calculate all prime factors of each number.
b. Create a column for each prime factor found in any of the integers.
c. Create a row for each integer.
d. In each cell of the table place prime factor raised to power. The power counts how many copies of the column’s prime factor were found for the row’s integer.

RESULT:
For GCF take lowest power of each prime factor found across all the integers. Count only the shared primes because the GCF is formed only out of the shared primes.
For LCM take highest power of each prime factor found across all the integers, not just the shared primes.

100: 2x2x5x5
140: 2x2x5x7
250: 2x5x5x5

Prime Column:

100 2^2 5^2
140 2^2 5 7
250 2 5^3

GCF = 2 x 5 = 10
LCM = 2^2 x 5^3 x 7 = 3,500

Once you become more familiar with calculating GCF and LCM using the prime columns you can just skip that part and just when you do the prime factorization scan the numbers and take the lowest powers of only the shared primes to find the GCF and highest powers of all occurring primes to find the LCM.

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7
Q

Determining Divisibility Through GCF:

A

GMAT may ask what combinations of numbers could lead to a specific GCF. When calculating the GCF for a set of numbers, you determine the prime factors and then take each prime factor all numbers have in common to the lowest power. If you are given the GCF, you work backwards to determine what the factor can tell you about divisibility.

E.g. if the task is to figure out whether z is divisible by 6 and you are told that the GCF of z and 12 are 3 you proceed to solve like this:

First you look at the prime factors of 12: 2x2x3
If both z and 12 have the GCF of 3 it means that z is divisible by 3 because it must have at least one prime factor of 3. At the same time this information also means that z is not divisible by 2 because if it was then the GCF would be a number divisible by 2. Since 12 has two 2s, this tells you that z can not have any factor of 2. For a number to be divisible by 6 it must be divisible by 3 and 2. Since z is not divisible by 2 it can’t be divisible by 6.

E.g. if your task is still to figure out whether z is divisible by 6 and you are told the GCF of z and 15 is 15 you know this:

The prime factors of 15 are: 3 and 5. Since the GCF of z is 15 you know that z has at least one 3 and one 5 as prime factor. But other than that you don’t know much else. You don’t know whether z is divisible by 2 which it would have to be in order to be divisible by 6.

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8
Q

Finding Value of Variables Through LCM:

A

If you are given the LCM and asked to find the value of variable you can use prime factorization again. Remember that the LCM of two or more integers is always at least as large as any of the integers. That means the variable cannot be larger than the LCM.

E.g. If the LCM of a and 12 is 36, what are the possible values of a?

First find prime factors of 36: 2x2x3x3
Find the prime factors of 12: 2x2x3
Since the LMC of 12 and a has two 2s and the LMC contains each prime factor to the power it appears the most and since 12 also has two 2s, you know that a could only either have two, one or zero 2s. a cannot contain more than two 2s because 12 contains two 2s and the LMC already counted in the most 2s between the factors of the two numbers. You also can see that the LMC contains two 3s. 12 however only contains one 3. That means that a must contain two 3s.

So a could have the values:
3x3 = 9 (zero 2s)
3x3x2 = 18 (one 2)
3x3x2x2 = 36

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9
Q

Counting Factors and Primes

A

GMAT could ask you to count factors of large numbers in different ways:

  1. How many different/unique, distinct prime factors?
    Strategy: count each repeated prime factor only ONCE.
    E.g. in 1,400 = 2x2x2x5x5x7, you would say three.
  2. How many total prime factors (length)? Would ask What is the length of 252?
    length of integer is defined as total number of primes that when multiplied equal that integer.
    Strategy: not asking for distinct factors but all, so count all together, could also add the exponents.
    E.g. in 1,400 = 2x2x2x5x5x7, you would say 6.
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10
Q

Counting all Factors of Numbers:

A

GMAT could ask you how many different factors a number has (all factors, not just primes)

Two Methods:

  1. Factor Pair Method:
    Especially good for smaller numbers. Factors always come in pairs. Process:
    Make table with two columns, Small and Large, start with 1 in Small and write 72 as its pair in Large (e.g. if we’re looking for factors of 72). Then go through the numbers, 2 in Small, 36 in Large etc. Do that until the number run into each other, which is the case after 8 with has the pair 9. The next number to test would be 9 and that once has pair 8 but you have that in the list already. At that point you can stop and count the number of factors you found.
  2. Using Prime factorization:
    The factor pair method is too cumbersome for large numbers. Prime factorization can shorten the process considerably. Key is to consider each distinct prime factor separately.
    a) factor number into primes. E.g. 2000 = 2^4 x 5^3
    b) consider prime factor 2: because there are four 2s, there are five possibilities for the number of 2’s in any factor of 2000: none, one, two, three, four.
    c) consider prime factor 5: because there are three 5s, there are four possibilities for the number of 5s in any factor of 2000: none, one, two, three.
    d) find the total number of factors: because there are five possible decisions for the 2-factor (whether it’s in zero time, one time, two times etc.) and four possible decisions for the 5-factor there are a total of 5x4 = 20 different factors.

General Rule: If a number has prime factorization a^x x b^y x c^z (where a, b, and c are all prime), the number has (x+1)(y+1)(z+1) different factors.

E.g. 9,450 = 2^1 x 3^3 x 5^2 x 7^1
so number has (1+1)(3+1)(2+1)(1+1) = 2 x 2 x 3 x 4 = 48 factors.

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11
Q

Factors of Perfect Squares and Cubes:

A

All perfect squares have an odd number of factors. That’s because factors come in pairs and in perfect squares there is one factor pair where the numbers are identical, it’s the square root.

E.g. in 36 two of the factors, 6x6, are identical.

Any number that is not a perfect square will never have an odd number of factors because the only way to arrive at a an odd number is to have a factor pair in which the two factors are equal.

For large numbers it would be too much work to use the factor-pair technique to proof that a number is a perfect square or has an odd number of factors. There’s a better technique:

Perfect squares are formed from the product of two copies of the same prime factors.

E.g. 90^2 = (2x3^2x5)(2x3^2x5) = 2^2 x 3^4 x 5^2

REMEMBER: The prime factorization of a perfect square contains only even powers of primes. And any number whose prime factorization contains only even powers of primes must be a perfect square. If a number’s prime factorization contains any odd powers of primes, the number is not a perfect square. Same counts for perfect cubes. Perfect cubes are formed from three identical sets of primes, so all of the powers of the primes are multiples of 3 in the factorization of the perfect cube.

E.g. 90^3 = (2x3^2x5)(2x3^2x5)(2x3^2x5) = 2^3 x 3^6 x 5^3

So you also know that the prime factors of k^3 are the prime factors of k in sets of three. So, if you for instance in a GMAT question are given that k^3 is divisible by some number and need to find out the least possible value for k you can look at the prime factorization of that number and you know three identical sets of that prime factorization make up the prime factorization of k^3 and then you take just one of those three sets and you know the least possible value for k. You work with a prime box in these questions.

E.g. If k^3 is divisible by 240, what is the least possible value of integer k?

  1. Prime factorization of 240: 2x2x2x2x3x5
  2. Write it down in a prime box. You know that cubes are three identical sets of prime factors so write
                                     k^3
                               /    |    \
                             /      |      \
                            k      k      k
                            2     2      2 
                            2
                            3
                            5

This is the prime factors given for k^3 through the question stem. The prime factors of k^3 must contain the prime factors of 240. Now there is a part of the box not filled out yet. You know that cubes contain three sets of identical primes so you can fill out the rest:

                                     k^3
                                  /    |    \
                                /      |      \
                               k      k      k

                               2     2       2 
                               2     2       2
                               3     3       3
                               5     5       5

So you know that k has 2, 2, 3, and 5 as prime factors and so k must be a multiple of 60, which is also the least possible value for k.

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12
Q

Factorials and Divisibility:

A

N! is product of all integers from 1 to N. So, it’s divisible by all integers from 1 to N. Therefore, N! is a multiple of all integers from 1 - N.

Similarly, 10! + 7 is a multiple of 7 because 10! is divisible by 7 and 7 is too. So, 10! + 7 is divisible by 7.

10! + 15 is a multiple of 15 and therefore divisible by 15 because 15 is divisible by 3 and 5 and so is 10!

10! + 11! is a multiple of any factor from 1 to 10 because every integer from 1 to 10 inclusive is a factor of both 10! and 11!

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13
Q

Advanced Remainders:

A

Remainders are often expressed as integers but can also be expressed as fractions and decimals.

E.g. three ways to express remainder when 17 is divided by 5:

17 = 3x5 + 2

  1. 17/5 = 3 with remainder of 2
  2. 17/5 = 3 2/5
  3. 17/5 = 3.4

So, if you are for instance given a decimal and asked to find a possible remainder all you have to do is set up the fraction.

E.g. what is a possible remainder when A/B = 4.35

You know remainder is 0.35 so just express that as fraction:

35/100 = 7/20

so remainder could be 7 or a multiple of 7 because you can change the fraction for instance to 14/40.

If you know for instance then that the remainder is expressed as 14/40 you can even find A/B:

4 14/40 = 160/40 + 14/40 = 174/40

I.e. A = 174, B = 40

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14
Q

0 as a Remainder in Quotients:

A

Remember that when you are told that 2 is a remainder when n is divided by 7 and you are supposed to find the possible values for n you just can look at multiples of 7 and add 2:

Possible values:

  • 0 because 0 can be quotient too and then 2/7 is the remainder.
  • 9 because 9/7 would leave a remainder of 2 and you calculate it by finding a multiple of 7 and adding 2.
  • 16 because 14 is the multiple of 7 and you added 2 because of the remainder given in question stem.

etc.

REMEMBER: in these questions you should not forget about the possible quotient of 0 which would only leave the fraction 2/7 and still fulfill the prerequisite that it’s a remainder of 0.

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15
Q

Multiples:

A

Multiples are the products of a given integer with other integers. An integer that is divisible by another integer without a remainder is a multiple of that integer. Multiples of an integer are either equal or larger than that integer.

E.g. multiples of 5: 5, 10, 15, 20, etc.
multiple of 3: 12 since 12 is divisible by 3.

               Multiple/Factor = Integer

REMEMBER: An integer is always both a factor and a multiple of itself. Every positive integer has infinite multiples.

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16
Q

Factors:

A

Factors, or divisors, of a number are the positive integers that divide into that number without a remainder. A factor of an integer is smaller or equal to that integer.

E.g. 36 has nine positive factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

                 Multiple/Factor = Integer

REMEMBER: An integer is always both a factor and a multiple of itself. 1 is a factor of every integer.

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17
Q

Greatest Common Factor/Greatest Common Divisor:

A

The greatest common factor, or greatest common divisor, of a pair of numbers is the largest factor shared by the two numbers.

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18
Q

Finding Common Multiples of Two Numbers:

A

E.g. Prime factorization is an efficient strategy here:

Prime factors of 4: 2 x 2
Prime factors of 6: 2 x 3

This shows us that we need a minimum of two 2s and one 3 to to “make” either a 4 or a 6 (not both at once). So, a number that is a multiple of both 4 and 6 is a multiple of:

2 x 2 x 3 = 12

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19
Q

Quick Way of Finding Multiples of Numbers:

A

If you are looking for multiples of 4 under 50 then divide 50 by 4. That gives you 12 and a remainder. That means that there are 12 multiples of 4 under 50.

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20
Q

Finding Least Common Multiple of Numbers:

A

You need to do prime factorization here to find the least common multiple of two numbers. Find the prime factors (prime numbers that divide an integer exactly) of the two numbers first.

E.g.
Prime factors of 10: 2 and 5
Prime factors of 12: 2, 2, and 3

Now combine the two to find the least common multiple of both 10 and 12:

2 x 2 x 3 x 5 = 60

That means 60 is the least common multiple of 10 and 12.
Now you know that multiples of 60 will also be multiples of 10 and 12, i.e. 120, 180, 240 are all multiples of 10 and 12.

Remember that if you are looking for a number that’s divisible by two specific numbers then you are essentially looking for the least common multiple of those numbers. If a question states that a number is divisible by two specific numbers and then gives you answer choices and asks you to determine which of these that same number is NOT necessarily divisible by, you are essentially asking, which one of these answer choices is not a factor of the common multiples of the two numbers. To make it easier we’ll look for the number in the answer choices that is not a factor of the least common multiple as that one is easy to determine. Do this:

  1. Determine the least common multiple.
  2. Look through the answer choices. Which ever number is not a factor of that common multiple, that common multiple is not divisible by.
21
Q

Remainder:

A

A remainder is what’s left over in a division problem. The remainder is always smaller than the number you’re dividing by.

E.g. 17 divided by 3 is 5 with a remainder of 2.

Remember: Most remainder problems on the GMAT are best solved by picking numbers.

Remember a remainder can be the numerator in a fraction.

E.g. in 7/5 = 5/5 + 2/5 which is a remainder of 2.

22
Q

Remainder of Numbers Divided by 10:

A

When a number is divided by 10, the remainder is simply the units digit of that number. For example, 256 divided by 10 has a remainder of 6.

23
Q

Questions asking for Divisibility:

A

Don’t forget, if a question asks if a number or equation is divisible by a number they are essentially asking whether if you do the division the result is an integer.

24
Q

Prime Numbers:

A

Prime numbers are integers that only have two positive factors, 1 and itself. 1 is not a prime number. There are an infinite number of prime numbers. There is no pattern to determine prime numbers.

2 is the first and the only even prime number.

Any positive integer with exactly two factors must be prime while positive integers with more than two factors are never prime. If a number x that’s ≥2 has only two factors then those factors must be x and 1 and x must be prime.

GMAT expects you to know prime numbers up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

25
Q

Prime Factorization:

A

Prime Factorization:

Means looking for all the prime numbers that a number can be divided by and multiplying them. For that it helps to write a factor tree. Prime numbers are all natural numbers greater than 1 that have no positive divisors other than 1 and itself.

Example: The prime factorization of 204.

First you look for the smallest prime that 204 can be divided by and that’s 2. 204 divided by 2 is 102. Smallest prime of 102 is again 2 and we get 51. The smallest prime of 51 is 3 and we get 17. 17 is a prime itself as well.

⇒ Prime factorization of 204 is 2x2x3x17.

REMEMBER very Important:
Prime Factorization is one of the most valuable tools for quant section on GMAT. Any question about multiples and factors is, at its heart, a question about prime factors. You often have to do the prime factorization to get to the solution. Prim factors can help you find all the factors of an integer. The factors can be found by building all the possible factors of the prime factors.

26
Q

Divisibility for 2:

A
  • Any even number is divisible by 2.
27
Q

Divisibility for 3:

A

A number is divisible by 3 if the sum of its digits is divisible by 3.

E.g. 4,317 is divisible by 3 because 15 is divisible by 3.

28
Q

Divisibility for 5:

A
  • Any number that ends in 0 or 5 is divisible by 5
29
Q

Divisibility for 6:

A
  • An integer is divisible by 6 if it’s both divisible by 3 and 2.

E.g. 48 is divisible by 6 because it’s divisible by 2 and by 3.

30
Q

Divisibility for 7:

A

A number is divisible by 7 if the difference between its units digit multiplied by 2 and the rest of the number is a multiple of 7.

E.g. 147 is divisible by 7 because 14 - (7 x 2) = 0 which is divisible by 7, meaning it is a multiple of 7.

E.g. 682 is not divisible by 7 because 68 - (2 x 2) = 64, which is not divisible by 7, so not a multiple of 7.

31
Q

Divisibility by 4:

A

A number is divisible by 4 if it’s divisible by 2 twice or if its last two digits compose a two-digit number that is itself divisible by 4.

E.g. 1,732 is divisible by 4 because 32 is divisible by 4.

32
Q

Divisibility by 8:

A

A number is divisible by 8 if it’s divisibly by 2 three times or if its last three digits compose a number that is itself divisible by 8.

E.g. 76,848 is divisible by 8 because 848 is divisible by 8.

33
Q

Divisibility by 9:

A

A number is divisible by 9 if the sum of its digits is divisible by 9.

E.g. 16,956 is divisible by 9 because 27 is divisible by 9.

34
Q

Divisibility by 10:

A

A number is divisible by 10 if its units digit is zero.

35
Q

Divisibility by 11:

A

Divisibility by 11 is tested regularly on GMAT’s tougher quant questions, so be prepared. A number is divisible by 11 if the difference between the sum of its odd-placed digits and the sum of its even-placed digits is divisible by 11.

E.g. 5,181 is divisible by 11 (meaning it’s a multiple of 11) because (5 + 8) - (1 + 1) = 11, which is divisible by 11, meaning it’s a multiple of 11.

E.g. 827 is not divisible by 11 because (8 + 7) - 2 = 13, which is not divisible by 11.

Bonus rule for 11: If the digit in the tens place of a three-digit number is equal to the sum of the digits in that number’s hundreds and units places, then that number is divisible by 11.

E.g. 253 is divisible by 11 because (2 + 3) = 5

And you can find out what the result would be if you divided that three-digit number by 11 (so the quotient of that number divided by 11) with this rule: 
The quotient (result of a division) of the number divided by 11 will be a two-digits number composed of the digits in the three-digits number's hundreds and units places.

E.g. for example quotient of 253 divided by 11 is 23 because 2 and 3 combine to the number 23.
That means: 253 = 11 x 23

CAUTION: the rule about counting the hundred and units places in a three-digit number and seeing if it equals 11 to determine divisibility cannot be used to rule out divisibility, only to rule it in.

E.g. 715 is divisible by 11 even though (7 + 5) is not 1. But here you can use the rule above to determine: (7 + 5) - 1 = 11 which is divisible by 11.

36
Q

Divisibility by 44:

A

A number is divisible by 44 if it passes the divisibility test for both 4 and 11, since 44 = 4 x 11.

E.g. 1,848 is divisible by 44 because 48 us divisible by 4 and (1 + 4) - (8 + 8) = -11 which is divisible by 11.

37
Q

Combining divisibility rules to test divisibility of numbers:

A

You ca combine divisibility rules to figure out whether a number is divisible by other numbers. But it only works when the separate numbers you are testing do not have any factors in common.

E.g. if you test divisibility by 6 by seeing if a number passes divisibility tests for 2 and 3, that is OK because 2 and 3 don’t have any factors in common.

38
Q

Testing Divisibility on GMAT:

A

GMAT may test divisibility the reverse way. E.g. tell you that a number has ones digit of 0 which tells you that the number must be divisible by 0. Or they tell you that the sum of the digits of x is equal to 21, which tells you that the number is divisible by 3 but not by 9.

39
Q

Divisibility and Addition/Subtraction:

A

If you add two multiples of 7, you get another multiple of 7.

E.g. 35 + 21 = 56
Because (5x7) + (3x7) = 7 (5 + 3) = 7x8

If you subtract two multiples of 7 you get another multiple of 7.

E.g. 35 - 21 = 14
Because (5x7) - (3x7) = 7 (5 - 3) = 7x2

This also means that if N is a divisor of x and of y then it’s also a divisor of x+y.

E.g. if 7 is a divisor of 35 and 21 then it’s also a divisor of 35+21=56.

40
Q

Factor Foundation Rule:

A

If a is a factor of b and b is a factor of c then a is a factor of c. In other words, any integer is divisible by all of its factors and also all factors of its factors.

E.g. if 72 is divisible by 12 then 72 is also divisibly by all the factors of 12 (1, 2, 3, 4, 6, 12)

In this way 12, 6, 4, and all other factors are part of the foundation of 72 for example. It’s all a foundation of blocks of factors.

This also means that an integer is only divisible by another integer if it can be formed by its prime factors. So one good way to see if a number is divisibly by another number is to write down its prime factors and see if the prime factors can form the divisor we’re testing because a number n has to be divisible by any number made up of the primes of numbers it’s divisible by.

E.g. is 27 a factor of 72?

72 = 2x2x2x3x3

27 = 3x3x3

But we only have two 3s so 72 is not divisible by 27 which is therefore not a factor of 72.

REMEMBER: If you are given a variable and know it’s divisible by two numbers and want to see if it’s divisible by a third number you can do that with prime factorization as stated above. so find prime factors of all three numbers and see if with the prime factors given for number 1 and 2 you can build number 3. If so, the unknown number, the variable, is divisibly by number 3. the prime factors of number one and two are the prime factors of the variable. HOWEVER: REMEMBER that when you count the prime numbers of the third number you are only counting the maximum amount of repeated variables necessary to form the two first numbers. Those are the only primes that are DEFINITELY primes of the third number.

E.g. If J is divisible by 12 and 10, is j divisible by 24?

12: 2x2x3
10: 2x5
24: 2x2x2x3

Here you can’t assume that j has three 2s, you can only assume that j has DEFINITELY two 2s because that’s the maximum amount of 2s necessary to build 12 and 10. So, we actually don’t know if j is divisible by 24 because we don’t know what other prime numbers j might have. This brings us to another point:

By finding the prime numbers of two divisors of a variable, you have only found part of the prime factors of the variable. The variable is unknown, it could have other prime numbers. This way you have at least found part of the prime numbers. But for example, you found that the third number was also a factor of the variable, which means some of the prime numbers of that third number might have to be added to the list of prime numbers of the variable (depending on whether or not you can form that third number with the prime numbers already in the prime box from the prime numbers from the other two integers). In this way there could be other prime factors of the variable. This also means that if you have a variable and you know it’s divisible by a number, e.g. 80, and you are asked if it’s also divisible by another number, e.g. 15, and you find that the prime numbers of 80 don’t let you form 15 it doesn’t mean for sure that the variable cannot be divided by 15 because with the prime numbers o 80 you were only given part of the prime numbers for the variable. It could have other variables and actually be divisible by 15. But in DS you have to be able to give a definitive answer and if you can’t then the information provided is not sufficient.

41
Q

Divisibility by 30:

A

If you’re asked if something is divisible by 30 remember that 30 can be factored into: 2x3x5. That means, the question can be rephrased as does the number have at least one factor each of 2, 3 and 5.

42
Q

Divisibility Questions:

A

Remember that in complicated divisibility questions you have to simplify the given expression as much as you can to bring it into a product form as much as possible (since divisibility has to do with the products of a number).

e.g. out of k(m^3-m) make:
km(m^2-1) = km(m-1)(m+1)

because the quadratic expression m^2 - 1 can be factored into: (m - 1) (m + 1)

43
Q

Divisibility in Consecutive Integers:

A

In a product of three consecutive integers you know that at least one number is even and at least one number is divisible by 3 (because every third number is divisible by 3). That means you know that that product of three consecutive integers will definitely be divisible by 2 and by 3.

44
Q

Sum of Digits and Divisibility:

A

If the sum of the digits of a number is a multiple of 3, then that number itself must be divisible by 3.

45
Q

Prime Factors:

A

REMEMBER:

if 10 has prime factors 2 and 5
and 15 has prime factors 3 and 5

you know that to make either 10 or 15 you need one 2, one 3 and one 5.

That tells you: 30 = 2x3x5

That means prime factors of the product 30 is the least amount of prime factors necessary to make either 10 or 15.

46
Q

In Questions about Prime Numbers:

A

Remember in questions about prime numbers to test the special case of 2, which is the smallest as well as only even prime number.

47
Q

Distinct Factors of a Number:

A

Distinct Factors of a Number:

Quick way of finding distinct factors of a number:

First find prime factorization of the number.

Example:
Prime factorization of 90 = 2 and we get 45 which is divisible by 3 meaning the smallest prime is 3. That leaves us with 15 of which the smallest prime is again 3. That leaves us with 5 which is a prime itself.

⇒ Prime factorization of 90 is 2x3x3x5.

Now you have to write this equation with exponents.

That means: 2^1x3^2x5^1

Now take the exponents and add +1 to each and multiply: (1+1)x(2+1)x(1+1) = 2x3x2= 12

⇒ The number 90 has 12 distinct factors.

48
Q

Prime Factorization in Dealing with Scary Exponents:

A

Prime factorization is often the key to dealing with scary exponents. If GMAT problem gives you this:

35^8 then you can be sure that this is the way to the solution:

35^8 = (5 x 7)^8 = 5^8 x 7^8

Use prime factorization when you see complicated exponents like this as it’s most likely how GMAT wants you to get to the solution.

49
Q

Product of Prime Numbers:

A

Since prime numbers are only divisible by 1 and themselves, the product of prime numbers cannot have any other prime factors either. You can use this to know that if you quickly have to choose an answer choice, you can eliminate those that have factors of other prime numbers in them.

E.g. Product of 19 x 17 x 13 x 11 cannot have any other prime factors so you can eliminate answer choices that do.

GMAT Prep (21 decks)

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