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form: 36^k - 5^m

Question

Among all the numbers representable as 36k - 5m (k and m are natural numbers) find the smallest. 
Prove that it is really the smallest.

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Among all the numbers representable as 36k - 5m (k and m are natural numbers) find the smallest. 
Prove that it is really the smallest.

 

 

You mean smallest natural?

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k=1 m=2 smallest 11.

Now to prove your answer is the smallest.

 

 

Last digit of result always is 1. We should prove it cannot be 1.

36k = 5m +1, left side is dividable by 4. Right side is not.

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I am missing how that proves k=1 and m=2 is the smallest

 

Powers of 36 have last digit of 6; powers of 5 have last digit of 5.

Their (positive) differences have a last digit of 1, so numbers other than 1 11 21 31 41 51 ... are excluded.

k=1 and m=2 gives 11 - at worst the second smallest result.

Since k-man showed that 1 must be exclude, 11 is smallest result.

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