Number Systems

alaynaik wrote:Dear All,

I request you to help me with the following query:

If a prime number p divides 735^4 - 735^3 + 735^2 - 735 and 735^3 + 736 then what is the remainder when 734 is divided by p?

Ans Options) 0,1,2,3,4

Thanks.

I'm guessing that there is a typo, because (735^4 - 735^3 + 735^2 - 735) and (735^3 + 736) share NO common factors - therefore we cannot have a prime number p that divides them BOTH.

But now - let's assume that you meant 735^3 + 735, then the 2 expressions share LOTS of prime factors (all of the factors of 735 in fact - and more): 2, 3, 5, 7, 17 and 15889. But all of these give DIFFERENT remainders when you divide 734 by them (734 has exactly 2 prime factors: 2 and 367).

The two expressions do happen to play together in an interesting way if you use the version I wrote (735^3 + 735). If you take the original expression (735^4 - 735^3 + 735^2 - 735) and group factor, you get the following:

(735^4 - 735^3) + (735^2 - 735)
735^3(735-1) + 735(735-1)
735^3(734) + 735(734)
734(735^3 + 735)

**hence my thought that the second expression you wrote should have been 735^3 + 735**

A better version of this problem would read:

If a prime number p divides 735^4 - 735^3 + 735^2 - 735 but DOES NOT divide 735^3 + 735 then what is the remainder when 734 is divided by p?

This actually does seem a bit more GMAT-like.

[spoiler]If you factor as I did above, you see that p divides 734(735^3 + 735) but DOES NOT divide the (735^3 + 735) part, so it must be that it divides into the 734 part. The remainder would be 0.[/spoiler]