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What is the remainder when the positive integer n is divided by the positive integer k, where k>1

(1) \(n=(k+1)^3= k^3 + 3k^2 + 3k + 1=k(k^2+3k+3)+1\) --> first term, \(k(k^2+3k+3)\), is obviously divisible by \(k\) and 1 divide by \(k\) yields the remainder of 1 (as \(k>1\)). Sufficient.

(2) \(k=5\). Know nothing about \(n\), hence insufficient.

bunuel's solution is certainly the quickest way to solve. (Of course, it would be (k+1)^3 = k^3 + 3k^2 + 3k + 1, and since each of the "k" terms is divisible by k, the remainder is 1).

But if this deduction/concept didn't jump out at you, you could also solve fairly quickly by picking numbers. If we pick different integer values for k and always end up with the same remainder for n/k, then we can trust that (1) is sufficient. On the other hand, if ever we get different remainder values, we know at once (1) is insufficient:

(1): let k=2. Then, n = (2+1)^3 = 27. 27/2 leaves a remainder of 1. ----let k=3. Then, n = (3+1)^3 = 64. 63 is clearly divisible by 3. Thus, 64/3 leaves a remainder of 1 again. ----let k=4. Then, n = (4+1)^3 = 125. 124 is clearly divisible by 4. Thus, 125/4 leaves a remainder of 1 again.

Re: What is the remainder when the positive integer n is divided [#permalink]

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11 Nov 2013, 09:04

JoyLibs wrote:

What is the remainder when the positive integer n is divided by the positive integer k, where k>1

(1) n= (k+1)^3 (2) k=5

(k+1)^3

Take any value of k(for eg 8) K= 8 ; 9^3/8 = (1^3)/9 = 1(because each value of 9/8 gives remainder 1.Hence 1 X 1 X 1) k = 2 ; 3^3/2 = (1^3)/2 = 1 etc.A sufficient

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03 Jan 2015, 09:47

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07 Aug 2016, 22:38

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