If p is a positive odd integer, what is the remainder when p is divided by 4?

1. When p is divided by 8, the remainder is 5.
2. p is the sum of the squares of two positive integers




you can always plug in a bunch of numbers until you've satisfied yourself that the statements are sufficient.
for (1), just find the first few numbers that give remainder 5 upon division by 8: 5, 13, 21, 29, 37, etc. all of these give remainders of 1 upon division by 4, so that's convincing enough. sufficient. (note: the gmat WILL NOT give problems on which a spurious pattern appears, only to be broken after the 40th or 50th number; if you see a pattern persist for 4-5 cases, you can take it on faith that the pattern persists indefinitely.)
for (2), you should make the same realization you made above: one of the numbers has to be odd and the other even. then just try a bunch of possibilities:
1^2 + 2^2 = 5
2^2 + 3^2 = 13
3^2 + 4^2 = 25, etc
1^2 + 4^2 = 17
2^2 + 5^2 = 29
3^2 + 6^2 = 45, etc
all these give a remainder of 1 upon division by 4. sufficient.