Can we have any n € N, for which 7^n ends with the digit zero?

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Solution

If 7ⁿ has to end with the digit 0 for any natural number n,

then it has to be divisible by 10.

ie. its prime factorisation should have the factors of both 2 and 5 [ since 10=2x5 ]

But we know that 7=7x1

By the fundamental theorem of arithmetics, we also know that there exist no other factors for 7.

Thus we conclude that 7ⁿ cannot end with the digit 0 for any natural number n.

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