What is the remainder of the division 2^56 by 7?

Such kind of remainder questions are almost always about the pattern recognition. Now, let's check:
2^1 divided by 7 yields the remainder of 2;
2^2 divided by 7 yields the remainder of 4;
2^3 divided by 7 yields the remainder of 1;

2^4 divided by 7 yields the remainder of 2;
2^5 divided by 7 yields the remainder of 4;
2^6 divided by 7 yields the remainder of 1;
...

As you can see the remainder repeats in pattern of 3: {2, 4, 1}. 2^56 will have the same reminder as 2^2 (divide 56 by 3 and find the remainder: 56=3*18+2, so it'll match with 2^2). Hence the remainder will be 4.

If it were 2^60 instead of 2^56 then as 60 has the remainder of 0 upon division by 3, then 2^60 would yield the same remainder as 2^3, so 1.

Questions to practice:
PS questions on remainders: search.php?search_id=tag&tag_id=199
DS questions on remainders: search.php?search_id=tag&tag_id=198

Also check theory on remainders: compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html

Hope it helps.