$17^3 = 4913$, and $7^3 = 343$, they share the same last digit.
$15^4 = 50625$, and $5^4 = 625$. Also the same last digit, the question is why do they share the same last digit?
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Have you heard of cyclicity? This is a phenomenon that motivated a lot of the origins of abstract algebra. – Rushabh Mehta Apr 19 '19 at 21:04
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Try doing this multiplication by hand and all will become clear :) – Jair Taylor Apr 19 '19 at 21:24
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Do you know modular arithmetic or congruences? – Bill Dubuque Apr 19 '19 at 21:53
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4Duplicate of Do the last digits of powers of a number $n$ follow the same cycle as the last digits of the number $n$'s last digit's powers? and surely many others. – Bill Dubuque Jun 03 '19 at 16:02
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See also Calculating last digit of a number using binomial theorem – Bill Dubuque Jun 03 '19 at 16:03
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Because if $l$ is the last digit of $a$, then $a$ cab be written as $10b+l$ for some natural $b$ and therefore, by the binomial formula,$$a^n=(10b+l)^n=\overbrace{10^nb^n+\binom n110^{n-1}b^{n-1}l+\cdots+\binom n{n-1}10bl^{n-1}}^{\text{multiple of }10}+l^n.$$
José Carlos Santos
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The operation of "taking the last digit" is like finding the remainder after division by $10$ -- or in math speak it is like taking the number modulo $10$. The great thing about modular arithmetic is that it respects all of the basic operation you typically like to, namely addition and multiplication (and therefore exponentiation). This means you can either exponentiate and then find the remainder, or you can find a remainder and then exponentiate, and you can even mix and match if your expression is really large. This in fact forms the basis of many modular exponentiation algorithms used in cryptography.
