Question

The sum of the third powers of the first 100 natural numbers will have a unit digit of?​

Answer 1

$\Huge {\orange {\bf {\underline {\underline {AnSwer}}}:-}}$

10 = 2×5, so to know about things modulo 10 (think "the last digit"), it is enough to know about them modulo 2 and modulo 5.

For all x, we have

x² ≡ x (mod 2)   and therefore    x⁵ ≡ x (mod 2)

By Fermat's Little Theorem, we have that for all x

x⁵ ≡ x (mod 5).

Therefore x⁵ ≡ x (mod 10) for all x.

Hence

1⁵ + 2⁵ + 3⁵ + ... + 100⁵ ≡ 1 + 2 + 3 + ... + 100

= 100 × 101 / 2 = 50 × 101 = 5050 ≡ 0 (mod 10).

So the units digit is 0.