A positive integer n yields a remainder of 7 when divided by 12. What is the remainder whe… — GRE General Test

A positive integer $n$ yields a remainder of $7$ when divided by $12$ . What is the remainder when the expression $5n + 3$ is divided by $6$ ?

The remainder when

5n + 3

is divided by

6

2

Since

n

leaves a remainder of

7

when divided by

12

, we can write

n = 12k + 7

for some integer

k \ge 0

. Substituting this into

5n + 3

yields

5(12k + 7) + 3 = 60k + 38

. Factoring out

6

gives

6(10k + 6) + 2

. Because

6(10k + 6)

is divisible by

6

, the remainder of the expression when divided by

6

2

Express

n

in terms of the division algorithm for divisor

12

n = 12k + 7

for some non-negative integer

k

An integer that leaves a remainder of

7

when divided by

12

can be represented as

12k + 7

Substitute the expression for

n

into

5n + 3

and simplify.

5(12k + 7) + 3 = 60k + 35 + 3 = 60k + 38

Algebraic expansion allows us to analyze the entire expression modulo

6

Determine the remainder of

60k + 38

when divided by

6

60k + 38 = 6(10k + 6) + 2

Since

60k + 36

is an exact multiple of

6

, the leftover term

2

is the remainder.

Properties of Integers and Remainder Arithmetic

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