If $1^{100}+2^{100}+3^{100}+4^{100}+5^{100}≡R(mod\, 5)$ then value of R is :

4

The correct answer is Option (4) → 4

$1^n≡1(mod\,5)∀n∈N$

$∴1^{100}≡1(mod\,5)$   ...(1)

$2^4≡1(mod\,5)$

$(2^4)^{25}≡(1)^{25}(mod\,5)$

$∴2^{100}≡1(mod\,5)$   ...(2)

$3^4≡1(mod\,5)$

$∴3^{100}≡1(mod\,5)$   ...(3)

$4^4≡1(mod\,5)$

$∴4^{100}≡1(mod\,5)$   ...(4)

$5^n≡0(mod\,5)∀n∈N$

$∴5^{100}≡0(mod\,5)$   ...(5)

From (1), (2), (3), (4) & (5)

$1^{100}+2^{100}+3^{100}+4^{100}+5^{100}≡1+1+1+1+0(mod\,5)$

$≡4(mod\,5)$