binomial theorem - Finding the last digit of certain number raised to any power -

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i'm trying find last digit of result of number raised power, using binomial theorem, not modulus or something. please explain me why last digit of number's unit number raised power same original number raised same power using binomial theorem. ex. xv^y = v^y

also, found out each integer each cyclicity , understand that. i'm confused since: 17^8 = 7^8 = 7^4 since 8 multiple of 4. why not 7^2 = 7^8 well? 8 multiple of 2.

it's because of last digit raising power several times , not power.

7^1=...7 <= 7^2=...9 7^3=...3 7^4=...1 7^5=...7 <= 7^6=...9 7^7=...3 7^8=...1 7^9=...7 <=

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