00:01
All right, so for this problem, we want to show that a positive integer is divisible by three, only if the sum of its digits is divisible by three.
00:10
And we're going to be using corollary two.
00:12
So i went ahead and wrote it out here.
00:16
These are going to be the little tricks we use to get to the end of this problem.
00:20
And so we'll just start with the fact that if we let b be any integer, it's going to be an integer that's greater than one, then we can write out any integer uniquely in the form, a k, b to the power of k, plus a k minus 1, 2 times b to the k minus 1, right all the way down to a 1 times b to the power of 1 plus a not and b to the power of 0.
00:59
And sometimes, of course, this 1 is left off in this last one, since b anything to the power of 0.
01:04
Is equal to 1 is also left off.
01:11
Okay, well, in this case, we have base 10.
01:13
So we want to replace b with 10.
01:16
So that would just give us a k to 10 to the power of k...