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m is a three digit integer such that when it

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m is a three digit integer such that when it [#permalink] New post 17 Mar 2016, 09:05
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m is a three digit integer such that when it is divided 5, the remainder is y and when it is divided by 7 ,the remainder is also y .If y is a positive integer , what is the smallest possible value of m?

This question is from Manhattan. It has solution, but i could not understand .
Any help?
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Re: m is a three digit integer such that when it [#permalink] New post 17 Mar 2016, 15:56
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The first thing that came to mind for me was: what's ANY number that both 5 and 7 go into with the same remainder? 35 works, right? Both when divided into 35 have remainder 0. So look for the smallest 3-digit integer both 5 and 7 divide into evenly.

Sidebar: we can see that 5 and 7 must go into any multiple of 35. If numbers divide evenly into a certain x, then they also divide evenly into multiples of that x.

So we just need the smallest 3-digit multiple of 35, which is 105. But wait, we need the remainder to be a positive integer. If 5 and 7 divide into 105 evenly, then they should both divide into 106 leaving a remainder of 1, since 106 is 105 + 1.

BTW, this question has very little resemblance to any ETS GRE question I've ever seen... Much harder than a normal GRE question. Don't feel bad if it stumped you!

This question is also an illustration (at least for me) that it's easier to derive a principle from an example than to recognize the principle right away. If you see the principle right away, great, but if you don't, trial and error illuminates the principle.
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Re: m is a three digit integer such that when it [#permalink] New post 25 Apr 2018, 07:32
So the solution to this question can be saught after by seeking that the multiple of 5 and 7 must have a multiple of smallest 3 digit which can have a POSITIVE remainder.

Therefore 7*5 =35
35*2 =105,105 goes in both 7 and 5 and hence we need to have 105+1 =106 which lead us to have remainder 1 which is a positive number.
Re: m is a three digit integer such that when it   [#permalink] 25 Apr 2018, 07:32
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