Note:

Proof:

1. Suppose that abc, which is equal to 100a + 10b + c, is a multiple of 37.
2. Thus 10000a + 1000b + 100c is a multiple of 37. (Multiply the previous number by 100.)
3. Thus 10a + 1000b + 100c is a multiple of 37. (Subtract 9990a, which is 270a × 37.)
4. Thus 10a + b + 100c is a multiple of 37. (Subtract 999b, which is 27b × 37.)
5. Thus cab is a multiple of 37.

The proof that bca is a multiple of 37 is similar.