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RayyanTanvir
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If x and y are positive integers and x/y has a remainder of 5, what is 01 Apr 2020, 09:55
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joyseychow
If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

A. 6
B. 12
C. 18
D. 24
E. 30

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This ques is from Man prep book. I don't understand why smallest value of x is 5 and not 11.
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When x is divided by y, the remainder is 5.
Hence, y must be greater than 5.
Therefore, minimum possible value of y is 6.

Now, possible values of x such that when divided by 6 we get a remainder of 5 are 5, 11, 17 etc. Hence, minimum possible value of x is 5.

Hence, minimum possible value of xy = 5*6 = 30
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If x and y are positive integers and x/y has a remainder of 5, what is 31 Jul 2023, 22:50
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Given, x/y has a remainder of 5
Therefore y>5, and y min= 6
since remainder is 5, x min = 5

therefore xy to be minimum
= x min * y min
= 5*6
=30
hence E
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If x and y are positive integers and x/y has a remainder of 5, what is 19 Sep 2023, 07:55
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Hey i feel the answer must be 24 since the question is asking for smallest possible value of xy if you look at the factors of 24; 8*3 gives us 24 let us consider x to be 8 and y to be 3

when you divide 8/3 we have the reminder as 5


please check
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If x and y are positive integers and x/y has a remainder of 5, what is 19 Sep 2023, 08:12
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srirang
Hey i feel the answer must be 24 since the question is asking for smallest possible value of xy if you look at the factors of 24; 8*3 gives us 24 let us consider x to be 8 and y to be 3

when you divide 8/3 we have the reminder as 5


please check
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The remainder must be less than the divisor. That is, 8 divided by 3 gives the reminder of 2, not 5: 8 = 2*3 + 2.
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If x and y are positive integers and x/y has a remainder of 5, what is 09 Jul 2025, 09:04
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Why can’t this be x=8 and y = 3 it is 24 and satisfy all conditions as well?


Bunuel
CoIIegeGrad09

\(x=qy+5\), not \(y+5\), and note that \(q\) can be zero. In this case \(x=0*y+5=5\), this is smallest value of \(x\).

Hey Bunuel....you lose me after the last part above... I still don't see how y=6. Could you explain step by step? or does anyone have a solution that doesn't involve algebra?


\(\frac{5}{y}\) gives remainder 5 means \(y>x\). For EVERY \(y\) more than \(x=5\), \(\frac{5}{y}\) will give remainder of \(5\): \(\frac{5}{6}\), \(\frac{5}{7}\), \(\frac{5}{16778}\), ... all these fractions have remainder of \(5\).

I thought \(\frac{5}{6}\), is equal to .8 with a remainder of 2?

THEORY:
If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). Where \(a\) is a dividend, \(d\) is a divisor, \(q\) is a quotient (\(\geq0\)) and \(r\) is a remainder.

Now consider the case when dividend \(a\) is less than divisor \(d\). For instance: what is the ramainder when \(a=7\) divided by \(d=9\):

\(a = qd + r\) --> \(7=q*9+r\), as \(q\geq0\) and \(0\leq{r}<d\) --> \(7=0*9+r=0*9+7\) --> \(r=remainder=7\).

Hence in ANY case when positive integer \(a\) is divided by positive integer \(d\) and given that \(a<d\), remainder will always be \(a\). Note here that EVERY GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

So: \(\frac{7}{9}\), remainder \(7\); \(\frac{11}{13}\), remainder \(11\); \(\frac{135}{136}\), remainder \(135\).



Back to our original question:
If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

A. 6
B. 12
C. 18
D. 24
E. 30

When we got that \(x=5\) and \(\frac{5}{y}\) gives remainder of \(5\), according to the above we can conclude that \(y\) must be more than \(x=5\). As we want to minimize the \(x*y\) we need least integer \(y\) which is more than \(x=5\). Thus \(y=6\) and \(xy=30\).

Check it:
\(5=0*6+5\) --> \(x=5\) dividend, \(y=6\) divisor, \(q=0\) quotient and \(r=5\) remainder.

Answer: E.

Hope it helps.
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If x and y are positive integers and x/y has a remainder of 5, what is Updated on: 09 Jul 2025, 11:26
Originally posted by Krunaal on 09 Jul 2025, 11:24.
Last edited by Krunaal on 09 Jul 2025, 11:26, edited 1 time in total.
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Isha2204
Why can’t this be x=8 and y = 3 it is 24 and satisfy all conditions as well?

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The remainder in that case will be 2, and not 5, as explained here: https://gmatclub.com/forum/if-x-and-y-a ... l#p3269119
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