Repdigit divisibility
How many digits long is the smallest repdigit (composed of 1s) that is divisible by 7? (Must have at least one 1 to be viable)
Hint: when testing for divisiblity by 7, you can remember the sequence: 1, 3, 2, 6, 4, 5. if 1x(1s digit) + 3x(10s digit) + 2x(100s digit) + 6x(1000s digit) + 4x(10000s digit) + 5x(100000s digit) is divisible by 7, then the number is divisible by 7. If the number has more than 6 digits, the sequence repeats (+ 1x(1000000s digit) + 3x(10000000s digit) and so on.)
The answer is 6.
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1 solution
Using the sequence given above, we can see that
1 + 3 + 2 + 6 + 4 + 5 = 21, which is divisible by 7
And checking the previous partial sums, we see
1 = 1 1+3 = 4 1+3+2 = 6 1+3+2+6 = 12 and 1+3+2+6+4 = 16
None of which are divisible by seven.
So the smallest repdigit(of 1s) divisible by 7 must be 111111 which has 6 digits.