When a positive integer n is divided by 12, the remainder is 7. What is the remainder when… — GMAT

When a positive integer $n$ is divided by $12$ , the remainder is $7$ . What is the remainder when $n^2 + 5n + 11$ is divided by $12$ ?

Answer: 11

Since

n

leaves a remainder of

7

when divided by

12

, we can substitute

n \equiv 7 \pmod{12}

directly into

n^2 + 5n + 11

. Evaluating

7^2 + 5(7) + 11

gives

49 + 35 + 11 = 95

. Dividing

95

12

gives

7

with a remainder of

11

Express

n

using modular arithmetic

n \equiv 7 \pmod{12}

A positive integer

n

that leaves a remainder of

7

when divided by

12

can be expressed as

n = 12k + 7

for some non-negative integer

k

Substitute

n \equiv 7 \pmod{12}

into the target expression

n^2 + 5n + 11 \equiv 7^2 + 5(7) + 11 \pmod{12}

By the algebraic properties of remainders (modular arithmetic), substituting the remainder

7

for

n

yields an equivalent remainder modulo

12

Evaluate the arithmetic sum

49 + 35 + 11 = 95

Computing

7^2 = 49

5 \times 7 = 35

, and adding

11

gives

95

Find the remainder of

95

modulo

12

95 = 12 \times 7 + 11 \implies 95 \equiv 11 \pmod{12}

Dividing

95

12

yields a quotient of

7

and a remainder of

11

Modular Arithmetic and Polynomial Remainders

Question