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 Minimum of an expression (Posted on 2023-09-12)
What is the smallest positive value of 36^k-5^l ?

k, l are positive integers.

Credit due: Victors Linis, Ottawa University

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 re: Solution | Comment 2 of 3 |
(In reply to Solution by Paul)

Nice approach, Paul, but you do not need to bother with mod 6. The simpler version of your excellent solution is:

Consider the expression mod 4 and 5

Mod 4: the k term is a multiple of 4 and the l lterm = 1 mod 4 (since 5 == 1 mod 4) and so the expression is -1 mod 4

Mod 5: the k term is 1 mod 5 (since 36 = 1 mod 5) and the l term is a multiple of 5 so the expression is 1 mod 5

There's exactly one possible residue mod LCM(4,5) = 20.  The expression must be 11 mod 20  since 11 = -1 mod 4 and +1 mod 5.

If it's also positive, that means the smallest value it can possibly have is 11. Fortunately, by inspection it's easy to see that when k = 1, l = 2, the value of 36 - 25 is indeed 11. Since we have a concrete example where the expression's value is 11 and proof that it can't be smaller, this is the minimum.

Edited on September 12, 2023, 5:42 pm
 Posted by Steve Herman on 2023-09-12 17:40:23

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