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# What is the remainder when n2 + 7 is divided by 8, where n is an odd p

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2457
What is the remainder when n2 + 7 is divided by 8, where n is an odd p  [#permalink]

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19 Dec 2018, 10:21
00:00

Difficulty:

5% (low)

Question Stats:

88% (00:39) correct 12% (00:34) wrong based on 90 sessions

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What is the remainder when $$n^2 + 7$$ is divided by 8, where n is an odd prime number?

A. 0
B. 2
C. 3
D. 4
E. 6

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Re: What is the remainder when n2 + 7 is divided by 8, where n is an odd p  [#permalink]

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19 Dec 2018, 11:26
EgmatQuantExpert wrote:
What is the remainder when $$n^2 + 7$$ is divided by 8, where n is an odd prime number?

A. 0
B. 2
C. 3
D. 4
E. 6

let n=3
3^2+7=16
16/8 gives 0 remainder
A
VP
Joined: 18 Aug 2017
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Location: India
Concentration: Sustainability, Marketing
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Re: What is the remainder when n2 + 7 is divided by 8, where n is an odd p  [#permalink]

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20 Dec 2018, 00:30
upon checking relation for n^2+7 ; at n= 1,3,5,7,11.. we can say that the remainder would always be 0

IMO A ..

EgmatQuantExpert wrote:
What is the remainder when $$n^2 + 7$$ is divided by 8, where n is an odd prime number?

A. 0
B. 2
C. 3
D. 4
E. 6

_________________

If you liked my solution then please give Kudos. Kudos encourage active discussions.

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2457
Re: What is the remainder when n2 + 7 is divided by 8, where n is an odd p  [#permalink]

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01 Jan 2019, 21:46

Solution

Given:
We are given that
• “n” is an odd prime number

To find:
We need to find,
• The remainder when $$n^2 + 7$$ is divided by 8

Approach and Working:
• If we observe, all the positive integers can be written in the form 4k or + 4k + 1 or 4k + 2 or 4k + 3, where k is a non-negative integer.
o Out of these, 4k and 4k + 2 cannot be prime numbers, since they are divisible by 2, for any value of k.

• Thus, all odd prime numbers must be of the form 4k + 1 or 4k + 3

So, $$n^2$$ will be $$(4k + 1)^2$$ or $$(4k + 3)^2$$
• And, $$n^2 + 7$$ = $$(16k^2 + 8k + 8)$$ or $$(16k^2 + 24k + 16)$$ = $$8(2k^2 + k + 1)$$ or $$8(2k^2 + 3k + 2)$$
• Thus, both these expressions are divisible by 8, for any value of k

Therefore, the remainder is 0

Hence the correct answer is Option A.

_________________

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Re: What is the remainder when n2 + 7 is divided by 8, where n is an odd p &nbs [#permalink] 01 Jan 2019, 21:46
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