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If 1 + 12 + 123 + 1234 + 12345 + 123456 + 1234567 + 12345678 + 123456789 + 1234567890 + 12345678901 + 123456789012 is congruent to n modulo n, where 0 <= n < 9, what is the value of n?

 Nov 21, 2021
 #1
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If a number is congruent to n modulo n, then it is congruent to (n-n) modulo n, which means it is divisible by n.

Now, (using a calculator because I'm lazy) adding all of these numbers up, we get 

1 + 12 + 123 + 1234 + 12345 + 123456 + 1234567 + 12345678 + 123456789 + 1234567890 + 12345678901 + 123456789012 = 137174210008. 

Now n must be a factor of 137174210008, and n must be less than 9, so testing, we get that only 2,4, and 8 divide 137174210008 and are less than 9, so \(\boxed{n=\{2,4, or \space 8\}}\)

 

Hope this was helpful :)

 Nov 21, 2021
edited by abcdefghijklmnopqrst  Nov 21, 2021

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