When the positive integer n is divided by 12, the remainder is 7. What is the remainder wh… — GRE General Test

When the positive integer $n$ is divided by $12$ , the remainder is $7$ . What is the remainder when $n^2 + 3n + 5$ is divided by $12$ ?

The remainder when

n^2 + 3n + 5

is divided by

12

3

The correct answer is obtained by replacing

n

with its remainder

7

modulo

12

in the expression

n^2 + 3n + 5

. Evaluating gives

7^2 + 3(7) + 5 = 49 + 21 + 5 = 75

. Dividing

75

12

yields

6

with a remainder of

3

Express

n

in terms of its quotient and remainder modulo

12

n = 12q + 7

for some non-negative integer

q

, meaning

n \equiv 7 \pmod{12}

By the division algorithm, any integer

n

leaving a remainder of

7

when divided by

12

can be represented as

12q + 7

Substitute

n \equiv 7 \pmod{12}

into the polynomial expression

n^2 + 3n + 5

n^2 + 3n + 5 \equiv 7^2 + 3(7) + 5 \pmod{12}

Properties of modular arithmetic allow direct substitution of equivalent remainders into polynomial expressions.

Evaluate the arithmetic expression.

7^2 + 3(7) + 5 = 49 + 21 + 5 = 75

Compute the total numeric value before finding the final remainder modulo

12

Divide the total sum by

12

to find the final remainder.

75 = 12 \times 6 + 3

, so the remainder is

3

A remainder modulo

12

must be an integer in the range from

0

11

Properties of Remainders and Modular Substitution

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