When the positive integer n is divided by 7, the remainder is 4. What is the remainder whe… — GRE General Test

When the positive integer $n$ is divided by $7$ , the remainder is $4$ . What is the remainder when $n + 15$ is divided by $7$ ?

Answer: 5

Adding 15 to

n

increases the remainder by 15. The original remainder is 4, so the new sum of remainders is

4 + 15 = 19

. Dividing 19 by 7 gives a quotient of 2 and a remainder of 5.

Express

n

algebraically based on the given remainder rule.

n = 7k + 4

for some non-negative integer

k

By the division algorithm, any integer divided by 7 with remainder 4 can be written as a multiple of 7 plus 4.

Substitute

n

into the expression

n + 15

n + 15 = 7k + 4 + 15 = 7k + 19

We need to find the remainder of this new quantity when divided by 7.

Extract the largest multiple of 7 from 19.

7k + 19 = 7k + 14 + 5 = 7(k + 2) + 5

Grouping multiples of 7 isolates the remaining constant term.

Identify the final remainder.

The remainder is 5.

7(k + 2)

is completely divisible by 7, leaving 5 as the remainder.

Remainder arithmetic and divisibility properties

Question