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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Vince Kotchian - GRE verbal expert.

https://vincekotchian.com