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Playing with bases


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Suppose n is an integer in base 10. Note that n2 ends with 0 if and only if n ends with 0. Now consider integers represented in base b, where 5 <= b <= 9. Determine for which base b (if any) the following statement is true:
For any integer n, n2 ends with 0 if and only if n ends with 0.

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i'm pretty sure that statement is true for any base.

base 5: 10^2 = 100. (5^2 = 25)

base 6: 10^2 = 100. (6^2 = 36)

etc.

a number will end in zero in a particular base if and only if it's evenly divisable by the base value. and squaring a number doesn't change its divisablty with respect to the base.

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i'm pretty sure that statement is true for any base.

base 5: 10^2 = 100. (5^2 = 25)

base 6: 10^2 = 100. (6^2 = 36)

etc.

a number will end in zero in a particular base if and only if it's evenly divisable by the base value. and squaring a number doesn't change its divisablty with respect to the base.

It's true that if you start with a number that ends with 0, its square will also end with 0. But the OP asks for a case of

if and only if, meaning that if you can find a number that ends with 0, while its (integer) square root does not end with 0, then the statement is not true for that base. In his post, vistaptb gave two such occurences.

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