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When integer m is divided by 13, the quotient is q and the r [#permalink]

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06 Apr 2009, 13:08

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When integer m is divided by 13, the quotient is q and the remainder is 2. When m is divided by 17, the remainder is also 2. What is the remainder when q is divided by 17?

When integer m is divided by 13, the quotient is q and the remainder is 2. When m is divided by 17, the remainder is also 2. What is the remainder when q is divided by 17?

A. 0 B. 2 C. 4 D. 9 E. 13

Detailed explanations please.

From the definition of quotients and remainders, we have:

m = 13q + 2 m = 17a + 2

(note that the quotient is different in the second case). So we have

13q + 2 = 17a + 2 13q = 17a

and since this equation involves only integers, the primes that divide the right side must divide the left, and vice versa. That is, q must be divisible by 17, and a must be divisible by 13. If q is divisible by 17, the remainder is zero when you divide q by 17.

Of course, if you can see that q = 17 is one possible value for q here, you can use that to get the answer of zero quickly as well. _________________

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Got 0 as well but am I right in thinking that 0 is another possible value of q?

Yes, perfectly correct - and that makes the question quite easy! _________________

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But, once I reduce the equation to 13q=17x, I am unable to make any deductions...can someone provide a clear explanation on how to use algebra to derive the values when we still have variables?

Careful here; if 13q = 17p, all you can say is that q is a multiple of 17, and that p is a multiple of 13. There is no way to find the actual value of q or p, and you certainly cannot be sure that q=17. It could be that q=34 and p=26, for example.

In general, if you see an equation like 13q = 17p, and if q and p are integers, then 13q and 17p are *the same number*. So they must have the same divisors. Since 17 is a divisor of 17p, it must be a divisor of 13q, so q must be divisible by 17.

Alternatively you can rewrite the equation as p = 13q/17, and since p is an integer, 13q/17 must be an integer, from which again we have that 13q is divisible by 17, so q is divisible by 17. _________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

Re: When integer m is divided by 13, the quotient is q and the r [#permalink]

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28 Oct 2014, 03:19

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: When integer m is divided by 13, the quotient is q and the r [#permalink]

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11 Aug 2015, 13:22

Expert's post

jaspreets wrote:

Hi stewart,

Can this be the alternative approach?

m=13q+2 2=13(0)+2 15=13(1)+2 28=13(2)+2

and

m=17p+2 2=17(0)+2 19=17(1)+2 36=17(2)+2

Thus we know p=q and that's 0, so 0/17 = 0.

Yes, you are correct in your approach. This question can be solved either by algebra as shown by Ian above or by plugging a few values as you have done. The trick here is to realise that you are finding a number that gives a remainder of 2 with both 13 and 17. _________________

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