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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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For a positive integer n, what is the remainder when n(n+1) is divided by 12?
1) n is divisible by 3.
2) n is divisible by 4.

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Re: For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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MathRevolution wrote:
For a positive integer n, what is the remainder when n(n+1) is divided by 12?
1) n is divisible by 3.
2) n is divisible by 4.

12 = 4*3

We need to know if n is divisible by 4 and 3.

Stat 1: Insufficient...since we don't know about n is divisible by 4.

Stat 2: Insufficient...since we don't know about n is divisible by 3.

Both : Sufficient...n is divisible by 4 and 3...i.e n ( n+1) is divisible by 12...
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Re: For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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MathRevolution wrote:
For a positive integer n, what is the remainder when n(n+1) is divided by 12?

1) n is divisible by 3.
2) n is divisible by 4.

Target question: What is the remainder when n(n+1) is divided by 12?

Statement 1: n is divisible by 3
Let's TEST some values.
There are several values of n that satisfy statement 1. Here are two:
Case a: n = 3, in which case n(n+1) = 3(3+1) = 12. Here, 12 divided by 12 leaves remainder 0
Case b: n = 6, in which case n(n+1) = 6(6+1) = 42. Here, 42 divided by 12 leaves remainder 6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is divisible by 4
There are several values of n that satisfy statement 2. Here are two:
Case a: n = 4, in which case n(n+1) = 4(4+1) = 20. Here, 20 divided by 12 leaves remainder 8
Case b: n = 8, in which case n(n+1) = 8(8+1) = 72. Here, 72 divided by 12 leaves remainder 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that n is divisible by 3
Statement 2 tells us that n is divisible by 4
COMBINED, we know that n is divisible by 12.
If n is divisible by 12, then we can be certain that (n)(n+1) is divisible by 12.
If (n)(n+1) is divisible by 12, then (n)(n+1) divided by 12 will leave remainder 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

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Re: For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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ques- For a positive integer n, what is the remainder when n(n+1) is divided by 12?
1) n is divisible by 3.
2) n is divisible by 4.

when N is divisible by 3, then N is a multiple of 3 (N can be 3,6,9,12...etc)
so this case can have both situations where N(N+1) is divisible by 12 and not divisible by 12.
For example- let N=3, thus 3*4 is divisible by 12, but if N=6 then 6*7 is not divisible by 12.
we are not certain what the remainder will be. It can either be Zero or any other number

this statement is NOT SUFFICIENT. Thus we can eliminate choices A and D. we are now left with choices B,C,E

following the same above approach for statement 2 when N is divisible by 4, we can have two situations where N(N+1) is divisible by 12 ( for N=8) but N(N+1) is not divisible by 12 (for N=4,16..etc). thus again we are not certain what will be the remainder. it can either be Zero or any other number

So this statement is also NOT SUFFICIENT. We can eliminate choice B. 2 choices are left-C,E

now if we are given N is divisible by both 3 and 4, then N is a multiple of 12. Thus, N(N+1) is divisible by 12. now we are certain about the remainder
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Re: For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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==> In the original condition, there is 1 variable (n), so D is highly likely to be the answer. In the case of 1), if =3, n(n+1)=12 with the remainder of 0,and if n=6, n(n+1)=42=12*3+6 with the remainder of 6, hence not sufficient. In the case of 2), if n=4, n(n+1)=20=12*1+8 with the remainder of 8, if n=8, n(n+1)=72=12*6 with the remainder of 0, hence not sufficient. Through 1) & 2), n=12, 24, 36… all have the remainder of 0, hence unique, and sufficient. C is the answer.
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Re: For a positive integer n, what is the remainder when n(n+1) is divided  [#permalink]

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_________________ Re: For a positive integer n, what is the remainder when n(n+1) is divided   [#permalink] 06 Jul 2018, 20:29
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