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The remainder, when a number n is divided by 6, is p. The remainder

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Bunuel
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The remainder, when a number n is divided by 6, is p. The remainder [#permalink] New post   03 Feb 2021, 01:15
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The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.


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chetan2u
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Re: The remainder, when a number n is divided by 6, is p. The remainder [#permalink] New post   03 Feb 2021, 05:54
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Bunuel wrote:
The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.


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n=6x+p.
If x is even, say x=2y, then the answer is No, as n=6(2y)+p=12y+p. Thus, p=q.
If x is odd, say x=2y-1, then n=6(2y-1)+p=12y-6+p=12(y-1)+6+p. Thus q=6+p, and ans is Yes as p<q.

(1) n is a positive number having 8 as a factor.
n=8a=6x+p
Say a=x=1, then x is odd and answer is YES. If n=8, then p=2, but q=8.
But say a=2 and x=2, then x is even and answer is NO. If n=16, then p=q=4
Insufficient

(2) n is a positive number having 6 as a factor.
n=6b=6x+p
This means that p=0.
x=1, then n=6. p=0, but q=6.
x=2, then n=12. p=q=0.
Insufficient

Combined
The number n is multiple of both 8 and 6 or multiple of LCM(6,8), that is 24.
So n=24z=6*(4z). Thus x is even.
Also divisible by 24 means divisible by all its factors : 1,2,3,4,6,8,12,24
remainder is 0 in each case, that is p=q
Sufficient

C
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Re: The remainder, when a number n is divided by 6, is p. The remainder [#permalink] New post   03 Feb 2021, 12:28
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The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

Stat1: n is a positive number having 8 as a factor.
n/6, remainder p = 2,4,0,2.. and n/12, remainder q = 8,4,0,8...., so, for p<q, not sufficient

Stat2: n is a positive number having 6 as a factor.
n/6, remainder p = 0. and n/12, remainder q = 6,0,6...., so, for p<q, not sufficient

Combining both, LCM (8,6)= 24
n/6, remainder p = 0. and n/12, remainder q = 0 so, p=q Sufficient

So, I think C. :)
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The remainder, when a number n is divided by 6, is p. The remainder [#permalink] New post   21 Oct 2023, 14:40
Bunuel wrote:
The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.


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GMATNinja KarishmaB Bunuel: Any quicker way to solve this problem? I took 3:32 mins!!

Okay:
One way would be to begin with statement 2, because "p" will always be 0. But I still have to do the 6 multiplication table and find the remainder pattern for n/12. (will take about a min)
In statement 1, I would still have to find the remainder pattern n = 8K (8k/6 and 8k/12) before I know statement 2 alone is insufficient.
I'll combine both statements then. It would still take me like 3 minutes to answer this question. How do I reduce the time?
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Re: The remainder, when a number n is divided by 6, is p. The remainder [#permalink] New post   23 Oct 2023, 04:06
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Engineer1 wrote:
Bunuel wrote:
The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q. Is p < q?

(1) n is a positive number having 8 as a factor.
(2) n is a positive number having 6 as a factor.


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GMATNinja KarishmaB Bunuel: Any quicker way to solve this problem? I took 3:32 mins!!

Okay:
One way would be to begin with statement 2, because "p" will always be 0. But I still have to do the 6 multiplication table and find the remainder pattern for n/12. (will take about a min)
In statement 1, I would still have to find the remainder pattern n = 8K (8k/6 and 8k/12) before I know statement 2 alone is insufficient.
I'll combine both statements then. It would still take me like 3 minutes to answer this question. How do I reduce the time?



Check this video on division and remainders first: https://youtu.be/A5abKfUBFSc

The remainder, when a number n is divided by 6, is p. The remainder, when the same number n is divided by 12, is q.
This tells me that EITHER q = p OR q = p + 6 (so p < q) depending on how many complete groups of 6 there are in n. If n has odd number of groups of 6, then p < q.
If n has even number of groups of 6, then p = q. So we need to find whether n has odd number of groups of 6 or even number of groups of 6 to find whether p < q or not.

(1) n is a positive number having 8 as a factor.

Here n can have odd number of groups of 6 (when n = 8) or even (when n = 16). Not sufficient.

(2) n is a positive number having 6 as a factor.

Again n can have odd number of groups of 6 (when n = 6) or even (when n = 12). Not sufficient.

Using both, we see that n has 8 and 6 as factors so it certainly has 24 as factor (LCM of 6 and 8). This means that n has an even number of groups of 6 and hence q = p
or simply, n is divisible by 24 means it is divisible by both 6 and 12 hence p = q = 0. Hence answer is "definite No".

Answer (C)
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Re: The remainder, when a number n is divided by 6, is p. The remainder [#permalink]
23 Oct 2023, 04:06
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