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Post subject: GMAT Number Theory Posted: Thu Jul 01, 2010 11:23 am 

Joined: Sun May 30, 2010 3:15 am Posts: 424

Positive integer n is divisible by 5. What is the remainder when dividing (n + 1)(n + 2)(n + 8)(n + 9) by 5? A. 0 B. 1 C. 2 D. 3 E. 4
(E) The fastest way to solve this question is to simply plug in n = 5.
We can do so because the question asks us to find the remainder for the formula that has n in it which can take many values. Therefore the remainder must be the SAME for ALL the possible values of n.
(n + 1)(n + 2)(n + 8)(n + 9) = (5 + 1)(5 + 2)(5 + 8)(5 + 9) = 6 × 7 × 13 × 14 = 7644. The remainder when dividing 7644 by 5 is 4. The right answer is choice (E).
If it wasn't GMAT and we needed to prove that the remainder is the same for any n divisible by 5 then the reasoning would be different.  Could you, please, provide a reasoning that the remainder is the same for any n divisible by 5?


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Gennadiy

Post subject: Re: math (test 1, question 37): number theory Posted: Thu Sep 02, 2010 3:13 am 

Joined: Sun May 30, 2010 2:23 am Posts: 498

We can factor (n + 1)(n + 2)(n + 8)(n + 9), then analyze the terms and find a remainder. But there is no need in lengthy factoring if we see that each term except 1 × 2 × 8 × 9 will contain n.
If term contains n then it is divisible by 5 and doesn't affect divisibility of the formula. Therefore the remainder of the formula (n + 1)(n + 2)(n + 8)(n + 9) when divided by 5 is same as when 1 × 2 × 8 × 9 is divided by 5. 1 × 2 × 8 × 9 = 144. The remainder is 4.
If it is not clear to you, you can do factoring all the way (simplifying all the brackets) or do some factoring in following way: (n + 1)(n + 2)(n + 8)(n + 9) = n(n + 2)(n + 8)(n + 9) + 1 × (n + 2)(n + 8)(n + 9). First term is divisible by n (and therefore by 5) so we can leave it the way it is  it doesn't affect the divisibility of overall formula.
Let’s do some factoring for second term: 1 × (n + 2)(n + 8)(n + 9) = n(n + 8)(n + 9) + 2 × (n + 8)(n + 9). Again the first term is divisible by n so we leave it the way it is and repeat the process couple more times: 2 × (n + 8)(n + 9) = 2 × n(n + 9) + 2 × 8 × (n + 9)
Once again we take the second term and simplify it: 2 × 8 × (n + 9) = 2 × 8 × n + 2 × 8 × 9
In result we get: (n + 1)(n + 2)(n + 8)(n + 9) = n(n + 2)(n + 8)(n + 9) + n(n + 8)(n + 9) + 2 × n(n + 9) + 2 × 8 × n + 1 × 2 × 8 × 9
All the terms that contain n would be divisible by 5. Therefore the remainder when dividing (n + 1)(n + 2)(n + 8)(n + 9) by 5 is the same as the remainder when dividing 1 × 2 × 8 × 9 by 5.
1 × 2 × 8 × 9 = 144. Therefore the remainder is 4 and the answer is (E).


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