More changes of base

If we change base 100 to base 101, we can readily verify divisibility by 17 (17*6=102) just as if we were using 11 on base 10 in decimal.

Moreover, if we change base 10 to base 11 and then to base 12, divisibility by 13 can be examined by the same token.

Examples: 5 30 57 (100) = 5 30 - 2*5 57 - 5 = 5 20 52 = 5 20 32 (101)

We just reverse-engineered the operation of adding groups of digits using the coefficients of the Pascal triangle (here 2 and 1, 1)

32 - 20 + 5 = 17, so the initial decimal number is divisible by 17.

289 (100) = 287 (101) = 2 87 (101). 87 - 2 = 85 = 5 * 17 so 289 is divisible by 17 (in fact, it's 17's square).

1001 (10) = A * A * A + 1 (11). A * A (11) = 91 (11). 91 * A (11) = 82A (11). 1001 (10) = 82A + 1 (11) = 830 (11). We see that the number is a mulltiple of the base, i.e. 11. 830 (11) = 8 * A1 (12) + 3*B (12) = 688 + 29 (12) = 6B5 (12). B5 + 6 = BB (12), divisible by 11 in base 12, which is 13 decimal.