What is the smallest positive value of 36^k-5^l ?
k, l are positive integers.
Credit due: Victors Linis, Ottawa University
(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
re: Solution
