How would you prove
1110-1 is divisible by 100?
Factoring is the clue.
Note that 1 to the tenth power is still one and then factor the left side.
1110 - 1 = 1110 - 110
1110 - 110 = 1110 - 110 = (11-1)(119 + 118 + 117 + 116 + 115 + 114 + 113 + 112 + 111 + 1)
Now note the right side. Clearly, the first factor is divisible by 10.
The second factor has a clever sum of its ten terms. The sum of ten integers each ending with the digit 1, is surely 0 in the ones (units) place. Thus, this factor also has to be divisible by 10.
Inasmuch as both factors on the right are divisible by 10, their product has to be divisible by 100. Therefore,
1110-1 is divisible by 100