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GMAT Quant > Properties of Numbers > Flashcards

Properties of Numbers Flashcards

1
Q

Integer definition

A

A number that can be written without a decimal or fraction. This includes 0 and negatives

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2
Q

Whole number definition

A

All nonnegative integers (this includes 0)

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3
Q

Is 0 an odd or even number?

A

It’s technically even

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4
Q

Is 1 a prime number?

A

No. The first prime number is 2

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5
Q

Represent even integers as an expression

A

2n (where n is an integer)

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6
Q

Represent odd integers as an expression

A

2n - 1 or 2n + 1 (where n is an integer)

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7
Q

When adding or subtracting 2 numbers how do we get an even number?

A

The sum or difference will only be even if both numbers are even or both numbers are odd.

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8
Q

When adding or subtracting 2 numbers how do we get an odd number?

A

One of the numbers must be odd and one of the numbers must be even to result in an odd sum or difference

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9
Q

What combination of odd and even numbers if multiplied result in an even product?

A

An even number multiplied by any integer (odd or even) will always be an even product

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10
Q

What combination of numbers if multiplied will result in an odd product?

A

If every number being multiplied is odd, then the product will also be odd

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11
Q

Even/Odd = ?

A

Even

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12
Q

odd/odd = ?

A

Odd

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13
Q

Even/Even = ?

A

Could be even or odd

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14
Q

What is absolute value?

A

The distance of a number from 0

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15
Q

What is a “signed number”?

A

A number that could be positive or negative

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16
Q

If x is raised to an even exponent will it result in a positive or negative number?

A

Positive

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17
Q

If x is raised to an odd exponent will it result in a positive or negative number?

A

It will have the same sign as the original base. If x is negative x^3 will be negative, if x is positive x^3 will be positive

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18
Q

What is another name for a factor?

A

A divisor

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19
Q

If y is a factor of x, then what 2 things must be true?

A
  1. x/y is an integer
  2. y is equal to or greater than 1 and less than or equal to x
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20
Q

If y is not 0, what must be true for x to be a multiple of y

A

x/y must be an integer

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21
Q

What are the first 25 prime numbers?

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

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22
Q

How does one determine the total number of positive integer factors of a number?

A

First, find the prime factorization of the number, making all exponents visible (even exponents of 1).
Second, add 1 to the value of each exponent and multiply the results. The product is the total number of factors.

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23
Q

How can I determine the number of unique prime factors of 9^12

A

When raised to a positive number, the number of unique prime factors remains the same (they’ll just be raised to a higher value exponent)

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24
Q

Define LCM?

A

LCM is the least common multiple. This is the smallest positive integer into which the given set of numbers will all divide into. Example: LCM of 5 and 2 is 10

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25
Q

How do I find the LCM?

A
  1. Find all the prime factors of the set of numbers
  2. If prime factors are repeated, take the one with the largest exponent
  3. Take all non-repeated prime factors
  4. Multiply the numbers taken aside in steps 2 and 3
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26
Q

If 2 positive integers do not share prime factors, what is the LCM?

A

The product of the 2 numbers

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27
Q

Define GCF?

A

GCF is the greatest common factor. This is the largest number that will divide into all of the numbers in a set. Example: GCF of 8, 12, 16 is 4

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28
Q

How do I find the GCF of a set of numbers?

A
  1. Find the prime factors of each number
  2. Identify repeated prime factors
  3. Of any repeated prime factors, take the numbers with the smallest exponent (no repeated prime factors means GCF is 1)
  4. Multiply the numbers from step 3 and GCF is the product
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29
Q

If y divides evenly into x, what is the LCM and what is the GCF?

A

GCF is y
LCM is x

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30
Q

If we know the LCM and GCF of x and y, what is the product of x and y?

A

LCM * GCF = xy

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31
Q

The LCM of a set of numbers will have all of the unique prime factors of the number set, so how can I find the unique prime factors of the product of the number set?

A

The unique prime factors will still be the same as unique prime factors of the LCM

32
Q

If light x flashes every 32 seconds and light y flashes every 12 seconds, how do I find when the lights will flash at the same time?

A

LCM of light x and y.

LCM is 96 so they blink at the same time once every 96 seconds

33
Q

What is the easiest way to solve 3,660/42?

A

Prime factorization. Once we have the prime factorization of the numerator and denominator, we can cancel things out and quickly find the answer.

34
Q

If x is divisible by y, what else is x divisible by?

A

Any factor of y

35
Q

If Z is divisible by 3 and 4, what else is Z divisible by?

A

12 (because 12 is the LCM of 3 and 4)

36
Q

How do I know if an integer is divisible by 3?

A

The sum of the digits is a number that is divisible by 3

37
Q

How do I know if an integer is divisible by 4?

A

The last 2 digits will be divisible by 4 (e.g. 244 or 516) or the number is a multiple of 100

38
Q

How do I know if an integer is divisible by 6?

A

The number will be divisible by 2 and 3, which means the number is even and the digits add to a multiple of 3.

39
Q

How do I know if a number is divisible by 7?

A

Just do the division, there is not an easy trick to this.

40
Q

How do I know if a number is divisible by 8?

A

The last 3 digits will be divisible by 8 (e.g. 2,240 or 3,400) or the number is a multiple of 1,000

41
Q

How do I know if a number is divisible by 9?

A

The sum of all of the digits is divisible by 9

42
Q

How do I know if a number is divisible by 11?

A

The sum of the odd places digits minus the sum of the even digits is divisible by 11. (Odd digits would be the 1st, 3rd, 5th…to the left of the decimal place. Remember that 0 is divisible by every number except itself!

43
Q

How do I know if a number is divisible by 12?

A

If a number is both divisible by 3 and 4

44
Q

What are consecutive numbers divisible by?

A

N! where N is the number of consecutive numbers. If there are 5 consecutive numbers (3,4,5,6,7) then it is divisible by 5!

It is also divisible by all factors of 5!

45
Q

n(n + 1), n(n - 1) is an example of what?

A

Representing 3 consecutive numbers algebraically. Consecutive numbers could be shown many ways such as (n + 1), (n + 2)

46
Q

(n - 1), (n + 1)

A

If n is odd, repeating even integers.

47
Q

What are n consecutive even integers divisible by?

A

(2^n) * n!

48
Q

What is the remainder of (12^127)/3 ?

A
  1. We know that 3 is a factor or 12. It goes into 12 evenly, so despite the large exponent the remainder is 0
49
Q

Express division as an equation (assume there is a remainder)

A

x/y = Q + r/y

x is the dividend (the numerator), an integer y is the divisor (the denominator), Q is the integer quotient of the division, and r is the nonnegative remainder of the division

50
Q

9.48 is 9 + what remainder?

A

9 + (many possible combinations)

Finding the smallest possible fraction will help us know the smallest possible remainder. The actual remainder will have to be a multiple of that

51
Q

If a number’s prime factors include 5 and 2, what does this say about the value of the number?

A

Each (52) pair is really 10, meaning the number of 52 pairs tells us how many 0s are at the end of a number. Additionally this tells us that each factorial that is greater or equal to 5! has a 0 at the end

52
Q

What can x/10, x/100, x/1,000 tell us about decimals?

A

The number of leading 0s. Example: 3/1000 is .003

53
Q

How do we determine the number of leading 0s for 1/x?

A

If X is not a perfect power of 10 then we can determine it by the number of digits in the denominator minus 1. Example: 1/1,001 will have 3 leading 0s

54
Q

Rewrite (5^5) * (2^5)

A

10^5

55
Q

How do I determine the number of leading 0s in the decimal 1/x when X is a perfect power of 10?

A

Leading 0s = the number of digits in the denominator minus 2

Example: 1/10 is .1 (no leading 0s)

56
Q

What is 0! ?

A

1

57
Q

What is n! divisible by?

A

1 to n inclusive and the product is divisible by any of the factor combinations of n!

58
Q

What numbers do perfect squares have to end with?

A

0, 1, 4, 5, 6, 9

59
Q

What is the rule regarding exponents of prime factors of perfect squares?

A

When doing prime factorization, the exponents of perfect squares will always be even

60
Q

What are the first 9 perfect cubes?

A

0, 1, 8, 27, 64, 125, 216, 343, 512

61
Q

What is a terminating decimal?

A

A decimal that ends

62
Q

How do I identify if a division problem will have a decimal that terminates?

A

The prime factorization of the denominator will contain only 2s, 5s, or both

63
Q

What is the remainder pattern when dividing consecutive numbers by n?

A

If n is the denominator, the remainder will begin at 0 and will be consecutive until n-1 before the pattern repeats

64
Q

How do I determine the units digit of the product of 88^5 * 99^6 * 77^3?

A

Use the exponent patterns to find the units digit of each number, multiply the units digits together, the units digit of the product of the 3 numbers is the answer

65
Q

How else can I write 3^ (16x+18)?

A

3^16x * 3^18

66
Q

What is the special remainder pattern when dividing numbers by 5?

A

The remainder will be the same for all numerators with the same units digit. Examples: all numbers with a 9 units digit have a remainder of 4, all numbers with a 7 in the units digit have a remainder of 2

67
Q

What is the greatest common factor of two consecutive numbers?

A
  1. This is because consecutive numbers never share any prime factors
68
Q

What is the exponent pattern when the units digit base is 2?

A

2, 4, 8, 6

69
Q

What is the exponent pattern when the units digit is 3?

A

3,9,7,1

70
Q

What is the exponent pattern when the units digit is 4?

A

4,6,4,6

71
Q

What is the exponent pattern when the units digit is 5?

A

5,5,5,5

72
Q

What is the exponent pattern when the units digit is 6?

A

6,6,6,6

73
Q

What is the exponent pattern when the units digit is 7?

A

7,9,3,1

74
Q

What is the exponent pattern when the units digit is 8?

A

8,4,2,6

75
Q

What is the exponent pattern when the units digit is 9?

A

9,1,9,1

76
Q

If 2^5, 3^3, and 13^2 are all factors of 936*W, how do I determine the smallest possible value of W?

A

Take the prime factorization of 936 and compare it against 2^5,3^3, and 13^2. The difference between those values and the prime factorization of 936 is the lowest possible value of W.

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