calculating mod

Question

17⁸ mod 47 =4

how do we calculate such huge values

Comments

$17^8\; mod \; 47 = 289^4\; mod\; 47 = (6*47 + 7)^4\;mod\;47 = 7^4\;mod\;47 $
$ 7^4 = 2401$

$\color{red}{2401\;mod\;47 = 4}$

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.==>7⁴mod47= 49²mod47 = 2 ² = 4

Answer 1

We can calculate this value using property of exponentiation in modular arithmetic

$If \ a \equiv b(modN), then\ a^{k} \equiv b^{k}(modN) \ for \ any \ positive \ integer$ k

$17^{8}(mod47) \\ \equiv (17^{2})^{4}(mod47) \\ \equiv 7^{4}(mod)47 \ \because 289mod47=7 \\ \equiv (7^{2})^{2}(mod47) \\ \equiv 2^{2}(mod47) \ \because \ 49mod47=2 \\ \equiv 4(mod47) \\ \equiv 4$