Hello all. I'm working on a proof and I'm needing a little help in the right direction. The statement to prove (via a direct proof) is:
For any positive integer n, n is a multiple of 9 if and only if the sum of the digits of n is a multiple of 9.
Here's what I have so far, but I think I'm making a mistake along the way:
Proof:
There exists a positive integer n that is a multiple of 9 if an d only if the sum of the digits of n is a multiple of 9.
(I will want to prove this using a direct proof)
If n is a multiple of 9, I can express this as 9n.
If the sum of the digits of n is a multiple of 9, I can express this as: 10k - 1.
Then the n in 9n can be substituted for 10k - 1. That will give us 9(10k - 1).
That is equal to 9(10k) - 9.
Both of these numbers are divisible by 9 ...
This doesn't seem like the correct way to do this proof. Help?
For any positive integer n, n is a multiple of 9 if and only if the sum of the digits of n is a multiple of 9.
Here's what I have so far, but I think I'm making a mistake along the way:
Proof:
There exists a positive integer n that is a multiple of 9 if an d only if the sum of the digits of n is a multiple of 9.
(I will want to prove this using a direct proof)
If n is a multiple of 9, I can express this as 9n.
If the sum of the digits of n is a multiple of 9, I can express this as: 10k - 1.
Then the n in 9n can be substituted for 10k - 1. That will give us 9(10k - 1).
That is equal to 9(10k) - 9.
Both of these numbers are divisible by 9 ...
This doesn't seem like the correct way to do this proof. Help?