Hello, Jenny!
Here's a more "primitive" proof . . .
Prove that every integer is congruent mod 9 to the sum of its digits.
We have an integer of the form:
Its value is: .
The sum of its digits is: .
Consider their difference:
. .
All the coefficients are of the form:
. . and hence are divisible by 9.
Since is a multiple of 9: .