Unless they are divided by themselves or one, they will always have a remainder.
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57 can be divided by 3 and 19 with no remainder. 51 can be divided by 3 and 17 with no remainder.
10
You know when you have a remainder once you have divided and all of the spots above the number being divided (the answer) have numbers. If you don't have zero by then, that last number is the remainder. If you want to figure it out without dividing, check a prime numbers chart.
17, 31, and 59
Yes, 7 is a prime number because it can only be divided by 1 and itself without leaving a remainder. There are no other whole numbers that can divide evenly into 7.