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Maelen's test > Number Properties: Divisibility & Primes > Flashcards

Number Properties: Divisibility & Primes Flashcards

1
Q

When should you use a prime box?

A

When the question asks whether or not a specific number is a factor of another number.

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2
Q 
definition of
 quotient
remainder
dividend
divisor
A

the number of times that the divisor goes into the dividend completely (always an integer)

the number that is left over if the dividend is not divisible by the divisor

the number being divided

the number doing the dividing.

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3
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 2?

A

2, if an integer is even

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4
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 3?

A

3, if the sum of the integer’s digits is divisible by 3

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5
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 4?

A

4, if the integer is divisible by 2 twice (48/2= 24, 24/2=12), or if the last two digits are divisible by 4. e.g. 23, 456

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6
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 5?

A

5, if the integer ends in 0 or 5

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7
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 6?

A

6, if the integer is divisible by both 2 and 3. e.g. 48 (divisible by 2 because it’s even, divisible by 3 because 4+8=12)

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8
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 8?

A

8, if the integer is divisible by 2, 3 times, or the last three integers are divisible by 8. e.g. 23,456, 456 is divisible by 2, 3 times and by 8

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9
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 9?

A

9, if the sum of the integers digits are divisible by 9

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10
Q

Rules of divisibility of certain integers:

What is the rule to help determine whether an integer is divisible by 10?

A

10, if integer ends in 0

all other numbers, ned to perform long division

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11
Q

How do you know a 3 digit integer is divisible by 11?

A

The first and hundredths place integers add up to the tenths place integer.

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12
Q

Find all factors of 72

A

factor pair chart

NPSG17.

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13
Q

What is this asking for:

“A positive integer that divides evenly into another integer. they are smaller than the original integer”

A

A Factor

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14
Q

What is this asking for:

A number formed by multiplying integer A by another integer. It is equal to or larger than integer A.

A

multiple of an integer A

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15
Q

N is a divisor of x and of y. is N a divisor of x-y?

A

Yes. Any multiples of any integer N, when you add or subtract them from each other, you get a multiple of N.

ex: if you add/subtract two multiples of 7 you will always get a multiple of 7
NPSG18

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16
Q

What is a prime number? What are the first 10 prime numbers?

A

positive integer that has no factors besides 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29

  • remember 1 is not prime bc it only has 1 factor.
  • remember 2 is the only even prime
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17
Q

if A is a factor of B and B is a factor of C, is A a factor of C?

A

Yes. Factor Foundation rule: any integer is divisible by all of its factors AND is divisible by all of the factors of it’s factors

EG: 72 is divisible by 12, it is also divisible by all the factors of 12 (1, 2,3,4,6,12).

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18
Q

Is 27 a factor of 72?

A

No.

Use a prime box: list all of the primes in a box for 72. Find the primes for 27. Are all of the primes of 27 in 72? No.

So 27 is not a factor.

  • When given a variable, you’ll create a partial prime box.
    e. g. given that the integer n is divisible by 8 and 15, is n divisible by 12? –> this means you can only create a partial prime box with 8 and 15’s factors, but there may be more.
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19
Q

2 x 5 x 6 x 10= 600

is 600 divisible by 8?

A

Yes, if you multiple integers by several even integers, the result will be divisible by higher and higher powers of 2 because each even number will contribute to at least one 2 to the factors of the product.

2 x 5 x (2x3) x (2x5) =600
8= 2 x 2 x 2
There are 3 two’s in 600, so it is divisible by 8
NPSG30

20
Q

sum of two primes:

when is it even and when is it odd?

A

Sum of two primes is always even. EXCEPT when one of the primes are 2.

(odd+odd=even)

  • if you see a sum of two primes that is odd. One of the primes is 2.
  • if you know that 2 cannot be one of the primes, then the sum is even.
21
Q

What are these asking for:

12 is divisible by 3
12 is a multiple of 3
12/3 is an integer
12 is equal to 3n where n is an integer
12 items can be shared among 3 people, so that each person has same number
3 is a divisor of 12 or 3 is a factor of 12
3 divides 12
12/3 yields a remainder of 0
3 goes into 12 evenly
A

All of these phrases are saying the same thing.

22
Q

what is the smallest prime number?

A

2

23
Q

what is the only even prime number?

A

2

24
Q

what are the first 10 prime numbers?

A

2, 3 ,5, 7, 11, 13, 17, 19, 23, 29

25
Q

The greatest possible factor of a and b is?

A

The positive distance between them. |a-b|.

Two numbers that are x distance apart may be multiples of that distance.
but cannot both be multiples of any number larger than x.

22-11=11 both can be multiples of 11 but they both can’t be multiples of any number larger than 11.

26
Q

How do you recognize divisibility and prime problems?

A

Problem states “factor” “multiple” “prime” or “divisible” “Is the positive integer n is a multiple of 24?”

27
Q

How do you find the GCF and LCM?

A

Find the prime factorization of each number.
Draw a ven diagram. In the overlap, write all of the common prime numbers. This is the GCF. no shared primes=1 in the shared space

The LCM is the product of all of the primes in the diagram. (shared area is only calc once)

28
Q

How do you find the GCF and LCM of more than 2 numbers?

A
  1. Calculate the prime factors of each integer
  2. create a columnn for each prime factor found within any of the integers
  3. create a row for each integer
  4. In each cell of the table, place a prime factor raised to a power. This power counts how many copies of the column’s prime factor appear in the prime box of the row’s integer.

To calculate the GCF: take the lowest count of each prime factor found across ALL integers.

To calculate the LCM: take the highest count of each prime factor found across ALL integers.

29
Q

What is the easiest way to solve:

How many different factors does 2000 have?

A
  1. Break 2000 into prime factors.
  2. Consider each DISTINCT prime separately.
  3. Consider 2 first: because prime far of 2000 contains (4) 2’s, there are 5 possibilities (with one possibility of no twos).
    * The number of possibilities for each distinct factor will= #of times the prime factor appears + 1.
  4. Once you have the number of possibilities for each factor. Multiply those two numbers to get the total amount of factors.
30
Q

What is special about the factors of perfect squares?

A

There is always an odd number of total factors.

Similarly, all integers that have an odd number of total factors MUST be a perfect square! All non square integers have an even amount.

31
Q

What is unique about prime factorization of perfect squares?

A

The prime factors only have even powers because a square would be twice the prime factors.
A number that has only even powers of primes in its factorization is therefore a perfect square.

16= 2^4

This extends to other “perfect” powers like cubes.
In cubes, all prime factors must be to a power that is a multiple of 3.

32
Q

If K^3 is divisible by 240, what is the least possible value of integer K?

A

60.

  1. Find the prime factors of 240. These factors must appear in k^3.
  2. Break down k^3 into three K’s.
  3. Distribute the known factors across the K’s. Any factors, not distributable go into the first k and are matched across the Ks.
  4. calculate value of one K, this is the least possible value.
33
Q

When positive integer A is divided by positive integer B the result is 4.35. Which of the following could be the remainder of A/B?

A

.35 refers to the remainder in relation to the divisor B. So B times .35 is the integer remainder. Since we don’t know B, we can calculate the fraction of the .35 which is 35/100 =7/20. We now know that the remainder must be a multiple of 7. Check Acs

34
Q

When a positive integer n is divided by 7 there is a remainder of 2. What is n?

A

Dividend=quotient X divisor + remainder

n= int x 7 + 2 create a chart for various integers.

This also means that N is 2 more than a multiple of 7.

35
Q

GMAT says: a gym class can be divided into 8 teams of equal number and divided into 12 teams with an equal number on each team.

A

total is multiple of 8 and multiple of 12

36
Q

GMAT says: integer m has an odd number of positive factors

A

Integer is a perfect square.

37
Q

How would you visualize “x is a multiple of 6”

A

visualize: x=6(int)

38
Q

x=27! +5

What is X a multiple of?

A

27!= 27x26x25x24….

Since there is a factor of 5 in 27! and you add 5, the sum is a multiple of 5.

39
Q

How do you visualize the divisibility of factors?

“6 is a factor of x”

A

X/6= int

factor tree or factor pairs

40
Q

Visualizing divisibility: how can you find all the prime factors of a number?

A

Prime factor tree or prime factor box

41
Q

GMAT says : 3^k

A

k= how many 3’s are there?

42
Q

GMAT: integer n has exactly two positive factors

A

N is prime

43
Q

GMAT says: x=3k+6 where k is an integer

A

x is a multiple of 3

44
Q

GMAT says: when n is divided by 7, the remainder is 0

A

N is a multiple of 7

45
Q

GMAT: n is an integer but (100+n)/n is not an integer

A

N is not a factor of 100.

46
Q

GMAT: x is 5 more than a multiple of 5.

A

x is a multiple of 5

47
Q

when is factoring an equation useful?

A

only when an equation is set to 0. xsq2 x 3x +4=0

Maelen's test (21 decks)

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