Yes, there’s a rule for every prime, although the rules get harder as the prime gets larger. Consider 7:

Since the following things are true:

1 leaves a remainder of 1 when divided by 7.

10 leaves a remainder of 3 when divided by 7.

100 leaves a remainder of 2 when divided by 7.

1,000 leaves a remainder of 6 when divided by 7.

10,000 leaves a remainder of 4 when divided by 7.

100,000 leaves a remainder of 5 when divided by 7.

1,000,000 leaves a remainder of 1 when divided by 7.

10,000,000 leaves a remainder of 3 when divided by 7.

100,000,000 leaves a remainder of 2 when divided by 7.

1,000,000,000 leaves a remainder of 6 when divided by 7.

This is repeating already (and you can show that it will continue to repeat the cycle 1,3,2,6,4,5). So what you do to tell if something is divisible by 7 is:

Multiply the one’s digit by 1. Find the remainder when divided by 7.

Multiply the ten’s digit by 3. Find the remainder when divided by 7.

Multiply the hundred’s digit by 2. Find the remainder when divided by 7.

Etc. (Continue to use the multipliers in the cycle 1,3,2,6,4,5.)

Now add up the remainders. Find the remainder of the sum when divided by 7. If the original number is divisible by 7, the remainder of the sum will be 0.