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GMAT: TTP Quant > Module 4: Properties of Numbers > Flashcards

Module 4: Properties of Numbers Flashcards

1
Q

Whole number (definition)

A

Non-negative integers. (All positive integers and zero)

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2
Q

Integer (definition)

A

A number that can be written without a decimal or fractional component. Includes positives, negatives, and zero

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3
Q

Zero is the only number that is… (3 things)

A

1) neither positive nor negative
2) equal to its opposite (0 = -0)
3) equal to all of its multiples

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4
Q

x^0 =

A

1 (ALWAYS)

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5
Q

Parity of zero

A

EVEN

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6
Q

The only number with exactly 1 factor is

A

1

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7
Q

2n +/- 1

A

Always an odd number. (2n is always even, +1 makes it always odd)

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8
Q

Parity with Addition/Subtraction of two values - integer rule

A

If the two values share the same parity, the result is even.

Even +/- Even = Even
Even +/- Odd = Odd
Odd +/- Odd = Even

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9
Q

Parity with Multiplication of two OR MORE values - integer rule

A

If ANY of the values are even, the result is even.

Even * Even = Even
Even * Odd = Even
Odd * Odd = Odd
Odd * Odd * Odd * Odd * Even = Even

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10
Q

Parity of Squares & Square Roots - integer rule

A

Always keep the parity of the value being squared/rooted

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11
Q

Parity with Division - integer rules (2)

A

Universal rules only apply if the denominator is odd.
Even / Odd = Even (ALWAYS)
Odd / Odd = Odd (ALWAYS)

No universal rule for:
Even / Even
Odd / Even (never divisible)

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12
Q

Remainders & parity when dividing a number by 2

A

Even / 2: No remainder

Odd / 2: Remainder of 1

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13
Q

Signed numbers (definition)

A

Positive and negative numbers. (Excluding 0)

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14
Q

Addition When Signs are Different

A

Positive + Positive = Larger Positive
Negative + Negative = Smaller Negative
Positive + Negative = Subtract smaller abs value from larger abs value. Keep the sign of the number with the greater abs value.

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15
Q

Subtraction when Signs are Different

A

1) If the first number & second number are both positive and the first number is greater, subtract normally.
2) If the above case is not true, change the subtraction of signed numbers to the addition of signed numbers.

(-6) - (-10) = -6 + 10 = 4

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16
Q

Factors & rule

A

The opposite of a multiple.

Given two whole numbers X and Y:

If Y divides evenly into X, Y is a factor of X.

x / y -> 16 / 4 -> 4 factor of 16

Rule: If a number is positive, the smallest factor is 1 and the largest factor is the number itself.

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17
Q

Factor Chart

A

Three columns: factor 1, factor 2, product. Determines all of the factors for number X, beginning with 1 and X. Count up from 1 in the Factor 1 column to find each factor. Stop counting once the factors “crossed” - this indicates that all factors have been found

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18
Q

Multiples (definition & 3 rules)

A

The product of number X and any integer.

1) For positive integers, the smallest multiple will always be 0 (x * 0 = 0.) The next smallest will be the number itself (x * 1 = x)
2) If X is a multiple of Y, then x = ny for some number N
3) If X is a multiple of Y, then x / y is an integer

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19
Q

Prime Numbers (definition)

A

Integers greater than 1 that have no factors other than 1 and itself

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20
Q

25 Prime Numbers less than 100

A 
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

(memorize the number of primes with each tens unit, in order: 4-4-2-2-3-2-2-3-2-1

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21
Q

Prime Factorization Tree (x Steps)

A

1) Using divisibility rules, attempt to break the number into two factors.
192: 1 + 9 + 2 = 12, so divisible by 3.

3, 64

2) Continue to break down remaining non-primes to the smallest primes possible.

3: 3
64: 2 * 2 * 2 * 2 * 2* 2

3) Re-write prime factorization using exponents

192 = (2)^6 * (3)^1

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22
Q

How to find the TOTAL number of factors in a particular number (2 steps)

A

1) Do prime factorization.
2) Add 1 to the value of each exponent in the prime factorization. Multiply those results. The resulting product is the total number of factors.

192 = (2)^6 * (3)^1
(6 + 1) * (1 + 1) = 7 * 2 = 192 has 14 factors.

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23
Q

Total Prime Factors vs. Unique Prime Factors (and 2 rules about unique prime factors)

A

Take 16 as an example:

  • 16 has 4 prime factors (2 x 2 x 2 x 2)
  • 16 only has 1 UNIQUE prime factor (they are all 2)

Rules:

1) If two numbers have the same unique prime factors, one is not necessarily divisible by the other.
2) The number of unique prime factors in a number does not change if the number is raised to a positive integer exponent.

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24
Q

Least Common Multiple, notes & how to find (2 methods)

A

Smallest positive integer into which all in a set of positive integers can divide from. It usually is not just the straight product of all numbers in the set.

Method 1:

1) Find prime factorization of each integer
2) For repeated prime factors in the set (appears 2x or more), only take those with the LARGEST exponent. (3^4, not 3^3)
3) Multiply all remaining prime factors. Result is LCM.

Method 2:
Take the largest integer of the set and test, in order, multiples of the integer until there is one that is divisible by all smaller integers in the set. (easier if numbers are simple.)

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25
Q

Greatest Common Factor & how to find (2 methods)

A

The largest number that will divide evenly into all numbers in a set.

Method 1:
1) Find prime factorization of each number
2) Identify repeated prime factors amongst numbers.
Take only the factors with the SMALLEST exponent.
4) If no prime factors are repeated, the GCF = 1.
5) Otherwise, multiply all of the REPEATED prime factors that were found. Result is GCF.

Method 2:
Work downward through the factors of the smallest number in the set until you find one that works. (easier if numbers are simple.)

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26
Q

When I see that X divides evenly into Y, I will

A

recognize that the smaller number X is the GCF of the two numbers, and the larger number Y is the LCM of the two numbers.

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27
Q

Finding the product of X and Y using their LCM & GCF

A

LCM(x,y) * GCF(x,y) = x * y

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28
Q

Things you can do with the LCM of a set of numbers (3)

A
  • Prime factorize the LCM to find all unique prime factors of a set of positive integers.
  • Determine when two processes that occur at differing rates or times will coincide.
  • If Z is divisible by both X and Y, Z must also be divisible by the LCM of X and Y.
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29
Q

Divisibility & Prime Factorization rules

A

When asked about divisibility, immediately think prime factorization.

1) If X is divisible by Y, then all prime factors of Y should have a corresponding factor in X. SIMPLIFY THE QUESTION STEM BY CANCELLING OUT CORRESPONDING FACTORS. This will reveal what is missing.
2) If X is divisible by Y, then X is also divisible by all factors of Y.

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30
Q

Divisibility with Exponents

A

When we divide like bases, keep the base and subtract the exponents.

3^3 / 3^2
3^(3-2)
3^1
3

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31
Q

Divisibility Rules: 0

A

No number is divisible by 0.

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32
Q

Divisibility Rules: 1

A

All numbers are divisible by 1.

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33
Q

Divisibility Rules: 2

A

All even numbers are divisible by 2.

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34
Q

Divisibility Rules: 3

A

If the sum of all digits in the number is divisible by 3, then the number is divisible by 3.

939
9 + 3 + 9 = 21
21 is divisible by 3
939 is divisible by 3

35
Q

Divisibility Rules: 4

A

3 rules:

1) If the last two digits of a number form a number divisible by 4, the bigger number is divisible by 4.

7,044 -> 44 -> yes

2) If the last two digits of a number are 00, the number is divisible by 4 (because 100 is a multiple of 4.)
3) If a number is divisible by two TWICE, then it is divisible by 4.

64/2 = 32/2 = 16
64 -> yes

36
Q

Divisibility Rules: 5

A

If the units digit of the number is 0 or 5, the number is divisible by 5.

37
Q

Divisibility Rules: 6

A

Even numbers whose digits sum to a multiple of 3 are divisible by 6. (must be divisible by 2 AND 3)

38
Q

Divisibility Rules: 7

A

Fuck you. There aren’t any easy ones. Use long division

39
Q

Divisibility Rules: 8

A

3 rules:

1) If a number is divisible by 2 THREE TIMES, then the number is divisible by 8.

64/2 = 32/2 = 16/2 = 8. YES

2) If the last three digits of an EVEN number are divisible by 8, the number is divisible by 8.

1160 -> 160/8 = 20 -> 1160 is divisible by 8

3) If the last three digits of a number are 000, the number is divisible by 8 (because 1000 is a multiple of 8.)

40
Q

Divisibility Rules: 9

A

If the sum of all of a number’s digits is divisible by 9, the number is divisible by 9.

479,655 -> 4+7+9+6+5+5 = 36 = yes

41
Q

Divisibility Rules: 10

A

If the ones digit is a zero, the number is divisible by 10.

42
Q

Divisibility Rules: 11

A

If the sum of the ODD-numbered place digits minus the sum of the EVEN-numbered place digits is divisible by 11, then the number is divisible by 11.

Odd places - Even places

253 -> (2 + 3) - 5 = 0 -> 0 is divisible by all numbers -> 253 is divisible by 11

43
Q

Divisibility Rules: 12

A

If a number is divisible by both 3 and 4, it is also divisible by 12.

Recap of each:

3: Sum of digits is divisible by 3
4: Last two digits form number divisible by 4 OR number is divisible by 2, twice OR number ends in 00

44
Q

Division will produce a remainder if ___ .

A

the numerator is NOT a multiple of the denominator

45
Q

Division with a Remainder (Formula)

A

x/y = Q + r/y

x = dividend (numerator)
y = divisor (denominator)
Q = integer quotient of the division
r = non-negative remainder of the division
46
Q

Distributive Property of Division

A

(x + y) / z = x/z + y/z

Works for addition and subtraction.

47
Q

Converting Remainders: Decimals to/from Fractions (3 Rules)

A

1) Fractions - You can always convert a fractional remainder to decimal form.
2) Decimals - If you DON’T KNOW the denominator: You CAN’T reliably determine a remainder from a decimal remainder. What you CAN do is determine what the most reduced fraction would be.

948 / 100 = 9.48 = 9 + 48/100
474 / 50 = 9.48 = 9 + 24/50
Most reduced: Just simplify 48/100 as much as possible (12/25)

3) Decimals - If you DO KNOW the denominator: Multiply the decimal component of the division by the divisor.

Given: 9 / 5 = 1.8
0.8 * 5 = 4 is the remainder.

48
Q

When simplifying remainders at the end of a multiplication, you must _____

A

correct excess remainders (where R > divisor)

49
Q

Finding remainder of a product of multiplication - steps

A

1) Find the remainder of each variable in the multiplication. (Use long division if needed.)

R of (500 * 600 * 700) / 8 ?
500/8 -> R4
600/8 -> R0
700/8 -> R4

2) Multiply the remainders together to find the remainder of the product.

404 = 0

3*) Correct excess remainders if necessary (where R > divisor)

50
Q

Adding & Subtracting remainders

A

1) Divide each term of the addition/subtraction, then either add or subtract the remainders. Correct excess remainders if necessary (where R > divisor)
2) (For subtraction) - the result may be negative, which is impossible. To correct this, add increments of the divisor to the result until it is positive - this is the true remainder.

51
Q

A remainder must always be ____ (3 things)

A

1) a NON-NEGATIVE (includes zero)
2) Integer
3) That is less than the divisor.

52
Q

Counting Trailing Zeros - 3 things to remember

A

1) Trailing zeros are created by pairs of (5 x 2). 1 pair = 1 trailing zero.

2) Scientific notation also indicates the number of trailing zeroes in a number:
52 x 100 = 5,200 = 52 x 10^2

3) Because of rule #1, any factorial >= 5! will have zero as its units digit (because there is one 5 x 2 pair embedded)

53
Q

Determine # of Digits in an Integer - 4 steps

A

1) Prime factorize the number(s)
2) Count the number of (5 x 2) pairs - each one = one trailing zero
3) Gather the number of unpaired 5s and 2s, along with other nonzero prime factors (if any), and multiply them all together. Count the number of digits in this product.
4) Sum steps 2 & 3.

54
Q

Determining Leading Zeros in Decimal Equivalent of a Fraction

A

If X is an integer with K digits:

Use form 1 / X

  • The result has k - 1 leading zeros
  • IF X is a perfect power of 10, then the result has k - 2 leading zeroes
55
Q

Divisibility of Factorials

A

RULE 1: n! is divisible by any integer from 1 to n, as well as the product of ANY factor combinations for numbers 1 through n.

1 - Simple example:
4! = 4 x 3 x 2 x 1 = 24
4! is divisible by 4 x 3 = 8

1 - Complex example:
10! = 10 x 9 x 8 … x 1
Need to prime factorize all numbers between 10 and 1 to find maximum combinations of each - then, use these thresholds to determine if another number can be divided by 10!

RULE 2: Any set of n consecutive positive integers is divisible by n! (n factorial)

10 * 11 * 12 * 13 is divisible by 4!

56
Q

Shortcut to Determine the Number of Primes in a Factorial (Using a Prime Number)

A

1) Divide the given factorial by the prime number in question, and record the quotient. Repeat this a few times, increasing the exponent of the prime each time. Stop once the division produces a quotient of zero.

Example:
21! / 3^n
21 / 3^1 = 7
21 / 3^2 = 21/9 = 2 (ignore the remainder)
21 / 3^3 = 21/27 = 0 (ignore the remainder)

2) Add the quotients.

7 + 2 = there are 9 factors of three in 21!
The largest value of N that would produce an integer = 9

57
Q

Shortcut to Determine the Number of Primes in a Factorial (Using a NON-Prime Number)

A

1) Break the non-prime value into prime factors.

“What is the largest integer value of n such that 40! / 6^n is an integer?”
40 / 6^n = 40 / 2^n * 3^n

2) Using the largest remaining prime value, do steps 1 & 2 from this process for PRIME numbers. (The largest prime will always be the limiting factor here.)

(Other steps: Divide the given factorial by the prime number in question, and record the quotient. Repeat this a few times, increasing the exponent of the prime each time. Stop once the division produces a quotient of zero. Sum the quotients. The resulting sum is your answer)

58
Q

Shortcut to Determine The Number of Primes in a Factorial (When The Base of the Divisor is A Power Of A Prime Number)

A

1) Where X is a non-prime number, translate it so it’s a factorial prime.

What is the largest integer of N such that 30! / 4n is an integer?

30! / 4n = 30! / 2^2n

2) Do the rest of the factorial shortcut.

30/2 = 15
30/4 = 7
30/8 = 3
30/16 = 1

15 + 7 + 3 + 1 = 26 2s in 30!.

3) Insert the power factorial into the numerator, drop the bases and create an inequality to find the largest possible value of N.

2^26 / 2^2n

2n <= 26
n <= 13

59
Q

Prime Factorization of Perfect Squares: a rule

A

Prime factorizations of a perfect square must contain only EVEN exponents.

100 = 10^2 = 5^2 * 10^2
64 = 2^6
60
Q

First 9 non-negative perfect cubes

A 
cube rt 0 = 0
cube rt 1 = 1
cube rt 8 = 2
cube rt 27 = 3
cube rt 64 = 4
cube rt 125 = 5
cube rt 216 = 6
cube rt 343 = 7
cube rt 512 = 8
61
Q

Prime Factorization of Perfect Cubes: a rule

A

Prime factorizations of a perfect cube must contain only exponents that are multiples of 3.

64 = 4^3 = 2^6
125 = 5^3
62
Q

Prime Factorization and Terminating Decimals: a rule

A

All fractions with denominators containing only 2s, 5s, or both produce decimals that terminate.

If any other prime factors exist in the denominator, then the decimal does not terminate.

63
Q

Remainder Patterns in Powers

A

Use remainder patterns to determine what the remainder of a number to a very large exponent will be - try raising the number into smaller powers (1, 2, 3, 4) and find the answer using the pattern that develops.

64
Q

Units Digit Positive # Power Pattern: Units digit = 2

A

Starting at 2^1, units digits follow the four-number pattern 2-4-8-6.

65
Q

Units Digit Positive # Power Pattern: Units digit = 3

A

Starting at 3^1, units digits follow the four-number pattern 3-9-7-1.

66
Q

Units Digit Positive # Power Pattern: Units digit = 4

A

Starting at 4^1, two number pattern: 4-6

4 = odd power
6 = even power
67
Q

Units Digit Positive # Power Pattern: Units digit = 5

A

All positive integer powers of 5 end in 5.

68
Q

Units Digit Positive # Power Pattern: Units digit = 6

A

All positive integer powers of 6 end in 6.

69
Q

Units Digit Positive # Power Pattern: Units digit = 7

A

Starting at 7^1, four number pattern: 7-9-3-1

70
Q

Units Digit Positive # Power Pattern: Units digit = 8

A

Starting at 8^1, four number pattern: 8-4-2-6

71
Q

Units Digit Positive # Power Pattern: Units digit = 9

A

Starting at 9^1, two number pattern: 9-1

9 = odd power
1 = even power
72
Q

Units Digit Positive # Power Pattern: Units digit = 0

A

All positive integer powers of 0 end in 0.

73
Q

Units Digit Positive # Power Pattern: Units digit = 1

A

All positive integer powers of 1 end in 1.

74
Q

How to determine where the Units Digit Positive # Power Pattern restarts

A

“What is the units digit of 7^49?”

1) Recall the power rule: for 7, the pattern is 7-9-3-1. Recognize that it’s a four-step pattern.
2) In an X-step power, powers of Y that are divisible by X will take the final units digit in the pattern. So, find the closest multiple of X that is less than the actual power, and count up from there.

Closest multiple of 4 to 49 is 48.
48 -> units digit = 1
49 -> units digit = 7

The units digit of 7^49 = 7

75
Q

When asked about the units digit of a product of complex variables, I will

A

remember that only units digits create units digits. You can ignore all other digits. Just multiply the units digits and take the units digit of that product.

Note: You can also use this shortcut for addition. In PS problems where each answer has a different units digit, all you need to determine is the units digit of each number and the solution units digit.

76
Q

Remainders after Division by Powers of 10

A

Division by 10: remainder = units digit of numerator
Division by 100: remainder = tens & units digit of numerator
Division by 1000: remainder = hundreds + tens + units of numerator

77
Q

Remainders after Division by 5: 1 rule

A

When integers with the same units digit are divided by 5, the remainder is constant.

7/5 = 1 2/5
17/5 = 3 2/5
78
Q

Common Evenly Spaced Set Patterns (3)

A

1) Consecutive integers [1, 2, 3, 4, 5]
2) Consecutive multiples of a given number [4, 8, 12, 16]
3) Consecutive numbers with a given remainder when divided by some integer

79
Q

Two consecutive integers will never ____. Therefore, _____

A

share the same prime factors.

Therefore, the GCF of two consecutive integers is always 1.

80
Q

When I see that (x + y) is an even number, I will

A

know that this means (x - y) is an odd number.

81
Q

Divisibility & Price Reduction problem - 3 steps.

A

“If the price of a computer were reduced by 16 percent, what could NOT be the final price of the computer?” (PS)

1) Recognize that the price in cents must be an integer. If F = final price and P = original price:

F = P(1 - .16)
F = .84P

2) Convert to a fraction and simplify

F = P(84/100)
F = P(21/25)

3) Invert the fraction and flip sides of the equation

(25/21)F = P

This tells us that F must be a multiple of 21, because 25 is not. Use prime factorization to solve

82
Q

Shortcut to Determine the Number of Primes in a Numerator with Factorial Addition/Subtraction

A

1) Factor the addition/subtraction in the factorial.

(43!) + (44!) = 43!(1 + 44) = 43!(45)
(31!) - (30!) = 30!(31 - 30) = 30!(30)

2) Determine the amount of X prime number in each term, separately (43! and 45)
3) Add the total. (Use inequality step to compare numerator & denominator and simplify - IF the denominator wasn’t a straight prime.)

83
Q

When I see a remainder problem, the first strategy I will consider is ______

A

Smart numbers. Often easiest

GMAT: TTP Quant (18 decks)

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