Divisible by 8?

Number Theory Level 3

$\large (102030405060504030201)_7$

What integer $0 \leq n \leq 7$ would need to be subtracted from the above number for it to be divisible by $(8)_{10}$ ?

Clarification:

No calculators, please!
The subscript 7 indicates that we are working in base 7.

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1 solution

Geoff Pilling
Apr 28, 2016

In base 7, the test for dividing evenly by 8 is to add up every other digit and subtract the total from the remaining digits, and if the result is divisible by 8, then the original number is. In this case, every other digit adds up to $1+2+3+4+5+6+5+4+3+2+1=36$ , and the sum of the other numbers is zero (since they are all zero). $36 \pmod 8 =4$ . So, if we subtract $\boxed4$ , the number will be divisible by $8_{10}$ .

Prove why the test works, and why testing for divisibility by $n+1$ in base $n$ is simple.

Alex G - 5 years, 1 month ago

This one is not as easy to prove, but it is similar to how you can test for divisibility by 11 in base 10. Lemme think about it for a bit...

Geoff Pilling - 5 years, 1 month ago

https://teresamccullough.wordpress.com/2011/05/25/divisibility-rules-for-numbers-in-bases-other-than-ten/

Brandon Huang - 1 year, 11 months ago

Divisible by 8?

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