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# M and N are integers. Find the remainder when is divided by

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SVP
Joined: 03 Feb 2003
Posts: 1607
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M and N are integers. Find the remainder when is divided by [#permalink]  27 Sep 2003, 10:58
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M and N are integers. Find the remainder when [M^2+N^2] is divided by 4.

(1) M and N are consequtive integers
(2) M is odd and N is even
Intern
Joined: 07 Sep 2003
Posts: 15
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D -

a. 2N^2 + 2N + 1 : odd
b. (odd)^2 + (even^2) : odd

both can't be divisible by 4.
SVP
Joined: 03 Feb 2003
Posts: 1607
Followers: 6

Kudos [?]: 84 [0], given: 0

The question is to find the remainder, and not to define whether the sum is divisible by 4 or not.
Intern
Joined: 29 Sep 2003
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Location: Dubai
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I would say A

'coz the sum of the squares of any 2 consecutive nos is 1 more than a multiple of 4.Hence no matter which 2 consecutive nos we take, we will always have 1 as the remainder

Egs:
2^2+3^3=13 which when divided by 4 gives 1 as the remainder
4^4+5^5=41 which when divided by 4 gives 1 as the remainder
Senior Manager
Joined: 21 Aug 2003
Posts: 258
Location: Bangalore
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Kudos [?]: 4 [0], given: 0

D both are suffecient.
1) Sum of two consecutive integers always give 1 as remainder.
2) (2m)^2 + (2n+1)^2 when M & N are even and odd
= 4(m^2 + n^2 + n) + 1
SVP
Joined: 03 Feb 2003
Posts: 1607
Followers: 6

Kudos [?]: 84 [0], given: 0

Vicky wrote:
D both are suffecient.
1) Sum of two consecutive integers always give 1 as remainder.
2) (2m)^2 + (2n+1)^2 when M & N are even and odd
= 4(m^2 + n^2 + n) + 1

D is correct. A small addition for (1) sum of SQUARES of two consequtive integers always gives 1 in a remainder when the sum is divided by 4. Lets prove it quickly.

If M and N are consequtive, then one is even and the other is odd
Let M=2k, and N=2k+1
M┬▓+N┬▓=4k┬▓+4k┬▓+4k+1=4(2k┬▓+k)+1
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