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Alexandra [31]
3 years ago
12

Prove that if a is a natural number, then there exist two unequal natural numbers k and l for which ak−al is divisible by 10.

Mathematics
2 answers:
EastWind [94]3 years ago
8 0
Assume a is not divisible by 10. (otherwise the problem is trivial). 
<span>Define R(m) to be the remainder of a^m when divided by 10. </span>
<span>R can take on one of 9 possible values, namely, 1,2,...,9. </span>
<span>Now, consider R(1),R(2),......R(10). At least 2 of them must have the sames value (by the Pigeonhole Principle), say R(i) = R(j) ( j>i ) </span>
<span>Then, a^j - a^i is divisible by 10.</span>
Elodia [21]3 years ago
8 0
Divisible by 10.........................................
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