McGraw-Hill Education GRE 2019

Since x is a multiple of 9, x must have two 3s in its prime factorization. Thus x =

3 × 3.... Since x is a multiple of 12, x must have two 2s and one 3 in its prime

factorization. You know that x already has a 3 in its prime factorization, so to

make x a multiple of 12, you only need to add two 2s to its prime factors. Thus

x = 3 × 3 × 2 × 2.... Since x is a multiple of 15, it must have one 3 and one 5 in

its prime factorization. You know from earlier that x already has a 3 in its prime

factorization, so to make x a multiple of 15, you only need to add one 5 to its prime

factors. Thus x = 3 × 3 × 2 × 2 × 5 = 180.

Remainders

Recall that 15 is a multiple of 5 since 5 divides evenly into 15. What about 16? 16 is

not a multiple of 5, because^165 = 3.2, which is a noninteger. When doing division,

if the numerator is not divisible by the denominator, the value left over after the

denominator divides into the numerator is called the remainder.

5 )

3

16

–15

1 Remainder

The remainder essentially tells you how many units the numerator is past a

given multiple of the denominator. It follows that the remainder must always be

smaller than the divisor. Look at the remainders yielded by each of the following:

4

4 = 1 r 0

5

4 = 1 r 1

6

4 = 1 r 2

7

4 = 1 r 3

8

4 = 2 r 0

Once you come to^84 , the remainder cycles back to zero.

Unknowns and Remainders

It helps to express remainders algebraically, especially when the numerator of the

fraction is an unknown. For example: When x is divided by 7, the remainder is 2.

You can translate this to mean: x is two units to the right of a multiple of 7.

7

9

14 21 28 35

16 23 30 37

As illustrated in the diagram, x can equal 9, 16, 23, 30, 37, and so on. There are

infinite values for x, but they are all two units to the right of a multiple of 7.

You can also express the previous mathematical sentence algebraically. Any given

multiple of 7 can be expressed as 7I, where I = any integer. Since x is two units to

the right of a multiple of 7, it follows that: x = 7I + 2.

CHAPTER 9 ■ NUMBER PROPERTIES 179

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