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I was looking at some proof questions and had difficulty answering a few of them How do I prove these statements below:

1) $3 \mid (10^{n+1} + 10^n + 1)$

2) $(a-b) \mid (a^n - b^n)$

user1176409
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gexcen
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  • Have you seen congruence? – Jearson Narvaez Rojas Oct 31 '14 at 06:11
  • I don't know if this will help you figure out a proof (probably not), but (a) is a special case of a useful fact you may have seen before: a number is divisible by $3$ if the sum of its digits (in base ten of course) is divisible by $3$. – bof Oct 31 '14 at 06:20
  • (1) $\space10=3\cdot3+1$ and (2) $\space (b)^n-b^n=0$ (otherwise it is an very known noteworthy product) – Piquito Jul 30 '23 at 13:42

1 Answers1

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Assuming all variables are integers, and $n\geq0$, since otherwise the questions do not make much sense.

In the first question use $10\equiv1\pmod3$ and compute the right hand side modulo$~3$. In the second question use $a\equiv b\pmod{a-b}$ and compute the right hand side modulo$~a-b$.

Marc van Leeuwen
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