When the positive integer n is divided by 12, the remainder is 7. Which of the following c… — GRE General Test

When the positive integer $n$ is divided by $12$ , the remainder is $7$ . Which of the following could be the remainder when $n^2 + 5n + 2$ is divided by $24$ ?

A
1
2Answer
C
5
D
7
E
11

Answer

Because

n \equiv 7 \pmod{12}

n

can be expressed as

12k + 7

. Testing the two cases for the integer

k

(even vs. odd) shows that

n \equiv 7 \pmod{24}

n \equiv 19 \pmod{24}

. Substituting

n = 19

into

n^2 + 5n + 2

yields

458

, and

458 = 19 \times 24 + 2

, which leaves a remainder of

2

. Therefore, the value

2

is a valid possible remainder.

Step-by-Step Solution

Express

n

algebraically based on the given remainder condition.

Since

n

leaves a remainder of

7

when divided by

12

n

can be written as

n = 12k + 7

for some non-negative integer

k

By the division algorithm, any integer leaving remainder

r

when divided by

m

is of the form

mk + r

Analyze

n

modulo

24

by considering the parity of

k

k

is even (

k = 2m

), then

n = 24m + 7 \equiv 7 \pmod{24}

. If

k

is odd (

k = 2m + 1

), then

n = 24m + 19 \equiv 19 \pmod{24}

To evaluate an expression modulo

24

when given information modulo

12

, split the integer into even and odd multiplier cases.

Evaluate

n^2 + 5n + 2 \pmod{24}

for both possible residue classes of

n \pmod{24}

Case 1 (

n \equiv 7 \pmod{24}

7^2 + 5(7) + 2 = 49 + 35 + 2 = 86

. Since

86 = 3 \times 24 + 14

, the remainder is

14

. Case 2 (

n \equiv 19 \pmod{24}

19^2 + 5(19) + 2 = 361 + 95 + 2 = 458

. Since

458 = 19 \times 24 + 2

, the remainder is

2

Substituting representative values of

n

into

n^2 + 5n + 2

determines all possible remainder outputs modulo

24

Identify which of the options matches a valid remainder.

The only possible remainders are

14

and

2

. Among the given choices,

2

is present.

Matching the derived valid remainder set

\{2, 14\}

against the options identifies the correct answer.

Key Concept

Modular Arithmetic and Remainder Properties under Quadratic Transformations

Estimated Time:2m 0s

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