HELP! probability that a 7 digit number is divisible by 7?

Modular Arithmetic

I don't know if this is correct but intuition tells me.

any 7 digit number can end with a digit consisting of 0 through 9.
Look at residual classes.

0 mod 7=0
1 mod 7=1
2 mod 7=2
3 mod 7=3
4 mod 7=4
5 mod 7=5
6 mod 7=6
7 mod 7=0
8 mod 7=1
9 mod 7=2

As there are 10 possibilities and two are residual class 0 , then 20% of these random seven digit numbers are divisible by 7.

Rap




Rap

Huh? This seems to be based on all 7-digit numbers - not the specific 7-digit numbers in the problem which don't contain the digits 0, 8, and 9 - and it doesn't make sense. 1/7th (give or take 1) of all N-digit numbers are divisible by 7. More generally, 1/7th (give or take 1) of any group of consecutive integers are divisible by 7.
Note that this fact doesn't get us closer to a solution. Is this for a programming class? If so, it's a trivial program to write. If not, I'm not sure how you'd reason out a solution.

Turns out to be 1/7.

Can you tell me why?

No - I wrote a program to calculate it. Is this for a class? If so, what subject and what topic within that subject?