Hi, what’s the source of these questions?

I’m completely stumped by both! (Didn’t really understand what the 2nd question is asking)

Appears to be GMAT. @pratik3818, you should cite the sources wherever possible.

For the first one, I would start by eliminating nonsensical answers. 64 is possible (8 and 8) and 72 is possible (9 and 8). There are no two integers from 1 to 10 that multiply to 66, 68 and 70 (try it; hint = factorise them).

Hence we can be sure there’s an 8 in it, and the next step would be to check whether the other digit is a 8 or a 9. There’s a nice hint though. Notice that \frac{25!}{15!} is divisible by 9 (because of the 18 and 21). Hence we can apply the divisibility test for 9:

1 + 1 + 8 + 6 + 1 + 6 + 7 + 6 + 2 + (A + B) = 2 + 15 + 21 + (A + B) = 38 + (A + B). This must be divisible by 9. This can happen when A + B = 16 (why?), and considering that one of A and B is 8, the other must be 8. Hence their product is 64.

For the second question, the question is poorly written to me but GMATClub has a solution to this you may wish to see. The idea is to find out which month is exactly the same as Jan 1991.

hey sorry, will cite now

when did i refused @sisi ?

I am confused about this part. so 25! includes 18 and 21 but how does that eliminate it from being divisible by 8? it also includes 16…

It is divisible by 8. The reason I looked for 9 instead is that it will allow us to make use of the divisibility test for 9. This allows us to quickly get the value of A + B, and the answer by extension.

Got it, Thanks!