When the positive integer m is divided by 8, the remainder is 5. What is the remainder whe… — GRE General Test

When the positive integer $m$ is divided by $8$ , the remainder is $5$ . What is the remainder when $3m + 7$ is divided by $8$ ?

The remainder is

6

Since

m

leaves a remainder of

5

when divided by

8

, we can evaluate

3m + 7

modulo

8

by replacing

m

with

5

. Calculating

3(5) + 7

gives

22

. Dividing

22

8

gives a quotient of

2

with a remainder of

6

because

22 = 8 \times 2 + 6

Express integer

m

using the division algorithm.

m = 8k + 5

for some non-negative integer

k

Dividing

m

8

yields a remainder of

5

Substitute the expression for

m

into

3m + 7

3m + 7 = 3(8k + 5) + 7 = 24k + 15 + 7 = 24k + 22

This expresses the new quantity algebraically in terms of

k

Determine the remainder when

24k + 22

is divided by

8

24k + 22 = 8(3k + 2) + 6

, so the remainder is

6

The term

8(3k + 2)

is a multiple of

8

, leaving

6

as the non-negative remainder less than

8

Remainder properties of linear combinations of integers

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