When the positive integer n is divided by 9, the remainder is 5. What is the remainder whe… — GRE General Test

When the positive integer $n$ is divided by $9$ , the remainder is $5$ . What is the remainder when $n^2 + 4n + 7$ is divided by $9$ ?

Answer: 7

Answer

The remainder when

n^2 + 4n + 7

is divided by

9

7

Any positive integer

n

that leaves a remainder of

5

when divided by

9

can be represented as

n = 9k + 5

for some non-negative integer

k

. Substituting this into

n^2 + 4n + 7

yields

(9k + 5)^2 + 4(9k + 5) + 7 = 81k^2 + 90k + 25 + 36k + 20 + 7 = 81k^2 + 126k + 52

. Since

81k^2

and

126k

are both divisible by

9

, the remainder of the entire expression when divided by

9

is determined entirely by

52

. Dividing

52

9

gives

52 = 9 \times 5 + 7

, so the final remainder is

7

Step-by-Step Solution

Express the integer

n

in terms of its remainder upon division by

9

n = 9k + 5

for some non-negative integer

k

, or equivalently

n \equiv 5 \pmod{9}

By the division algorithm, any integer divided by

9

can be written as a multiple of

9

plus the remainder.

Substitute

n \equiv 5 \pmod{9}

into the polynomial expression

n^2 + 4n + 7

n^2 + 4n + 7 \equiv 5^2 + 4(5) + 7 = 25 + 20 + 7 = 52 \pmod{9}

Properties of modular arithmetic allow substitution of remainder values into polynomial expressions.

Find the remainder of

52

when divided by

9

52 = 9 \times 5 + 7

, which gives a remainder of

7

Dividing

52

9

yields a quotient of

5

and a remainder of

7

, which is strictly between

0

and

8

Key Concept

Properties of Integer Remainders and Modular Substitution

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic