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M=O+P+Q, where O, P, and Q are consecutive positive integer; [#permalink]

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12 Sep 2006, 20:09

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M=O+P+Q, where O, P, and Q are consecutive positive integer; M=R*S*T, where R, S and T are positive consecutive integers. What is the remainder when M is divided by 5?
1). When O is divided by 5, the remainder is 1
2). When R is divided by 5, the remainder is 1

Q: If O, P, and Q are consecutive positive integer, is it safe to assume O < P < Q ?

it does not matter whether they are consecutively increasing or not.

I think it matters.
Lets say if O = 6 , P = 5 and Q = 7 then remainder is 3
if O = 6 P = 7 and Q = 8 then remainder is 1.

But I think it is safe to assume O<P<Q.

St1:
Let O = x then
M = x+x+1 + x+2 = 3x+3
x/5 has a remainder of 5. 3x will have a remainder of 3 and 3x+3 will have a remainder of 1: SUFF

St2: Let R = x
Then M = x(x+1)(x+2) = x^3 + 3x^2 + 2x
x/5 has a remainder of 1. This means last digit of x is either 1 or 6.
So last digit of x^3 will be either 6 or 1. i.e Remainder = 1
Last digit of x^2 will be either 6 or 1. Last digit of 3x^2 will be either 8 or 3. i.e remainder = 3
Last digit of 2x will be 2 - i.e remainder = 2

Total remainder = 1+3+2 = 6
Hence final remainder will be 6-5 = 1: SUFF

I found a solution here
C it is
(1) alone we have O divided by 5 remainder is 1 so O,P,Q divided by 5 must have remainder is one of these sets (0,1,2);(1,2,3);(4,0,1).So M diveded by 5 could have remainder of 3 or 1(1+2+3=6) or 0. This tells us nothing
(2) alone R,S,T divided by 5 must have remainder is one of these sets (0,1,2);(1,2,3);(4,0,1). So M diveded by 5 could have remainder of 2 or 1 or 4 . This also tells nothing
But(1)(2) together we can say that M divided by 5 must have remainder of 1