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# What is the remainder when the positive integer n is divided by 6?

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What is the remainder when the positive integer n is divided by 6?  [#permalink]

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03 Dec 2017, 01:04
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25% (medium)

Question Stats:

81% (01:05) correct 19% (02:04) wrong based on 47 sessions

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What is the remainder when the positive integer n is divided by 6?

(1) n when divided by 12 leaves a remainder of 1
(2) n when divided by 3 leaves a remainder of 1

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Re: What is the remainder when the positive integer n is divided by 6?  [#permalink]

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03 Dec 2017, 10:07
Bunuel wrote:
What is the remainder when the positive integer n is divided by 6?

(1) n when divided by 12 leaves a remainder of 1
(2) n when divided by 3 leaves a remainder of 1

(1) n/12 = 12q + 1/12: so n could be {1,13,25,37...} and the numbers in this set divided by 6 all give a remainder of 1. sufficient.
(2) n/3 = 3q + 1/3: so n could be {1,4,7,10,13...} and the numbers in this set divided by 6 give a remainder of 1 or 4. not suf.
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Re: What is the remainder when the positive integer n is divided by 6?  [#permalink]

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03 Dec 2017, 10:20
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Bunuel wrote:
What is the remainder when the positive integer n is divided by 6?

(1) n when divided by 12 leaves a remainder of 1
(2) n when divided by 3 leaves a remainder of 1

Target question: What is the remainder when the positive integer n is divided by 6?

Statement 1: n when divided by 12 leaves a remainder of 1
USEFUL RULE #1: "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

No onto the question...........
Statement 1 does not tell us the quotient, so let's just say that n divided by 12 equals k with remainder 1
So, we can write: n = 12k + 1, where k is some integer
We can also write 12k a different way: n = (6)(2)(k) + 1
Or n = (6)(2k) + 1
As you can see, (6)(2k) is a multiple of 6, which means (6)(2k) + 1 is ONE MORE than a multiple of 6
So, when (6)(2k) + 1 (aka n) is divided by 6, the remainder must be 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n when divided by 3 leaves a remainder of 1
USEFUL RULE #2: If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, for statement 2, some possible values of n are: 1, 4, 7, 10, 13, 16, 19, 22, etc.
Let's TEST some values...
Case a: If n = 1, then the remainder is 1, when n is divided by 6
Case b: If n = 4, then the remainder is 4, when n is divided by 6
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

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Re: What is the remainder when the positive integer n is divided by 6?  [#permalink]

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04 Dec 2017, 00:38
From 1) we know $$\frac{n}{12} = x + \frac{1}{12}$$ so adjusting it to become $$n = 12x + 1$$ if we let $$x = 0$$ then$$n = 1$$, plug in in the question stem $$\frac{1}{6}$$ remainder is 1, if we let $$x = 1$$, then $$n = 13$$, plug in $$\frac{13}{6} = 2 \frac{1}{6}$$ another remainder of 1 and so it goes. A is sufficient.

From 2) $$\frac{n}{3} = x + \frac{1}{3}$$ so adjusting it to become $$n = 3x + 1$$ if we let $$x = 0$$ then $$n = 1$$, plug in in the question stem $$\frac{1}{6}$$ remainder is 1, if we let $$x = 1$$, then $$n = 4$$, plug in question stem $$\frac{4}{6}$$ a remainder of 4. Because we get two different answers it is insufficient.

Re: What is the remainder when the positive integer n is divided by 6? &nbs [#permalink] 04 Dec 2017, 00:38
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