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 $n + 18$ is divided by $7$ ?

Answer: 1

The remainder when

n + 18

is divided by

7

1

Because

n

leaves a remainder of

4

when divided by

7

n

can be represented as

7k + 4

. Adding

18

gives

n + 18 = 7k + 22

. Since

7k

is a multiple of

7

, the remainder of

(7k + 22) \div 7

depends entirely on

22 \div 7

. Dividing

22

7

gives a quotient of

3

with a remainder of

1

Represent the positive integer

n

algebraically based on the given remainder.

n = 7k + 4

for some non-negative integer

k

By the division algorithm, any integer

n

divided by

7

with remainder

4

can be written as

7k + 4

Add

18

n

n + 18 = (7k + 4) + 18 = 7k + 22

Substitute

7k + 4

for

n

in the expression

n + 18

Determine the remainder when

7k + 22

is divided by

7

Since

7k

is divisible by

7

, the remainder is

22 \pmod 7 = 1

Dividing

22

7

yields a quotient of

3

and a remainder of

1

(

22 = 7 \times 3 + 1

Remainder Properties under Addition

Question