Knowing and understanding divisibility tests is an important mathematical skill. It is not enough to know how to do the tests -- knowing why the tests work is even more important. When you understand why a test works, you can recreate the test even if you have forgotten the details.
Here's one way to test if a number -- let's say, 55,762 -- is divisible by 7:
Remove the last digit, multiply it by 2, and subtract that from the revised number:
Repeat until you're left with a number that is either a multiple of 7 or not a multiple of 7:
Forty-nine is a multiple of 7. Therefore, the number 55,762 is divisible by 7. Notice that this test looks at least as long, if not longer, than actually dividing the number by 7. Still, it is interesting to do these tests, because it tells us more about how numbers relate to one another.