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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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k=1 m=2 smallest 11.

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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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it does not need to be natural, only k and M have to.

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But there can't be a smallest if you're including integers. Right?

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it does not need to be natural, only k and M have to.

k=1 m=-infinite (though -infinite is not real number)

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Natural numbers are positive or non-negative

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I see the error in my clarification. Yes the result should be natural. That is my mistake.

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k=1 m=2 smallest 11.

Now to prove your answer is the smallest.

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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

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k=m=0. Result is 0.
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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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