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# If p is a positive odd integer, what is the remainder when p

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If p is a positive odd integer, what is the remainder when p [#permalink]  16 Jan 2007, 09:17
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If p is a positive odd integer, what is the remainder when p is divided by 4?

(1) When p is divided by 8, the remainder is 5
(2) p is the sum of the squares of two positive integers
Senior Manager
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I would pick A

S1:When p is divided by 8, the remainder is 5. Then p=x8+5 where x8 is divisible by 4 as x8=x*4*2. So 5/4 gives a remainder of 1

S2:p=a^2+b^2 depends on a and b so insuff
Intern
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A.
bcos :

with a, (any number /8) having remainder 5 will always have 1 when dividing by 4.

for b., ( 9 + 25 )/4 remainder 2.
(4+9)/4 remained 1
so cant be determined with b.
Manager
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In A, the remainder will always be 1 using numbers such as 13, 21, 29, 85, etc.

In B, A and B could be anything so you dont know
Using 3 and 5 > 9+25 = 34 with remainder of 2
Using 2 and 3 > 4+9 = 13 with remainder of 1
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wall street...bulls, bears, people from connecticut

Senior Manager
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sorry guys.... you got it all wrong...

A is sufficient as everybody noticed....

but B is sufficient as well... here goes:
the important part is that p is odd (!!!) as given in the stem. so if it is a sum of two squares one must be odd and the other must be even....

but - an even square is always divisible by 4 ( since (2k)^2=4k^2), and an odd square always leaves a remainder of 1 when divided by 4 since (2k+1)^2=4k^2+4k+1
hence their sum (which is p) has a remainder 1 when divided by 4.
hence sufficient.

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Hobbit Excellent explanation. I got this wrong in GMAT Prep.
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Regards

Subhen

Senior Manager
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I was too careless to note that p is an odd integer. Can't afford to make mistakes like this.

Thanks hobbit.
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