n^4 is divisible by 32

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abhi332: Master | Next Rank: 500 Posts; Posts: 117; Joined: Wed Dec 30, 2009 1:57 pmFollowed by: 1 member; Location: India; Thanked: 4 times

by abhi332 » Thu Feb 25, 2010 12:48 pm

If n is an integer and n^4 is divisible by 32, which of the following could be the remainder when n is divided by 32?
(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

[spoiler]OA:B[/spoiler]

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Source: Beat The GMAT — Problem Solving | Thu Feb 25, 2010 12:48 pm

ajith: Legendary Member; Posts: 1275; Joined: Thu Sep 21, 2006 11:13 pmFollowed by: 2 members; Location: Arabian Sea; Thanked: 125 times

by ajith » Thu Feb 25, 2010 1:46 pm

abhi332 wrote:If n is an integer and n^4 is divisible by 32, which of the following could be the remainder when n is divided by 32?
(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

[spoiler]OA:B[/spoiler]

n^4 is divisible by 2^5
n^4 will be divisible by 2^8 also

n will be divisible by 4

[spoiler]If n is divisible by 4 it can leave only multiples of 4 as the remainder when divided by 32
In the list only B is a multiple of 4 hence B[/spoiler]

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analyst218: Master | Next Rank: 500 Posts; Posts: 140; Joined: Fri Feb 05, 2010 2:43 pm; Thanked: 3 times; GMAT Score:720

by analyst218 » Thu Feb 25, 2010 4:50 pm

hmm.
can you explain how you got to n is divisible by 4?

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Thu Feb 25, 2010 7:25 pm

n^4 is divisible by 2^5. Or, in other words, n^4 has at least five 2s in its prime factorization.

Since the question asks about n, let's think about what this means for n's divisibility by 2:

If integer n were divisible by only one 2, then we know that n^4 would be divisible by 2^4 but not by 2^5. So, in order for n^4 to be divisible by 2^5, n itself must have at least two 2s in its prime factorization. In other words, we can deduce that n is a multiple of 4.

Now, the question becomes: what could the remainder be when we divide a multiple of 4 by 32?

At this point, either we could use the rule that the poster above outlined or we can pick numbers:

32 is a multiple of 4, and 32 divided by 32 leaves a remainder of 0, which doesn't appear in the answer choices. So, the next multiple of 4, namely 36, when divided by 32 leaves a remainder of 4. So the remainder COULD be 4. Since the question is asking for what the remainder COULD be, and since there can only be one correct answer, choose B.

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abhi332: Master | Next Rank: 500 Posts; Posts: 117; Joined: Wed Dec 30, 2009 1:57 pmFollowed by: 1 member; Location: India; Thanked: 4 times

by abhi332 » Fri Feb 26, 2010 1:14 am

Hi Testluv,

n is the multiple of 4, so n can be 4, 8 16....

if suppose n is 8 and I divide it by 32 will leave remainder 8 only.

I may be missing something..

What you think, you become.

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ajith: Legendary Member; Posts: 1275; Joined: Thu Sep 21, 2006 11:13 pmFollowed by: 2 members; Location: Arabian Sea; Thanked: 125 times

by ajith » Fri Feb 26, 2010 1:15 am

abhi332 wrote:Hi Testluv,

n is the multiple of 4, so n can be 4, 8 16....

if suppose n is 8 and I divide it by 32 will leave remainder 8 only.

I may be missing something..

Abhi, do 8,12,16 etc feature the answers?

Always borrow money from a pessimist, he doesn't expect to be paid back.

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abhi332: Master | Next Rank: 500 Posts; Posts: 117; Joined: Wed Dec 30, 2009 1:57 pmFollowed by: 1 member; Location: India; Thanked: 4 times

by abhi332 » Fri Feb 26, 2010 1:21 am

Ok, got it , only 4 is in the answer choice which is multiple of 4..

What you think, you become.

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