Is there a quick way to tell which numbers are a product of consecutive integers?
The answer is: yes—with mental math! You should know multiplication tables up to 12, so, with the exception of 331, you should know which integers are products of which other integers right away.
For instance, you see 30 and think "hmmm, this is 5 times 6" and you see 72 and think "this is 8 times 9."
100 is 10 times 10, and you can't do 9 times anything or 11 times anything so it's not a product of consecutive integers. In general, a perfect square can't also be a product of two consecutive integers.
A product of consecutive integers must be even (because one of the two consecutive numbers will be even), so 21, 45, and 331 can't work.
Regarding mental math in general:
The GRE allows you to use a calculator, but using a calculator for every single problem will only slow you down. The more you exercise your mental math skills, the better! Besides multiplication tables up to at least 12, you should know how to convert fractions into percents/decimals without any problem. You should know the square of every number up to 20 and the cubes for every number up to ten. Be able to approximate calculations like 27 x 14 in a matter of seconds. Doing so will help immensely on test day! And we you see something trickier—then it's time for the calculator!