The remainder when 3^{37} is divided by 80 is | vedclass

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Question

(C) We need to find the remainder of $3^{37}$ when divided by $80$.We can write $3^{37} = 3 	imes (3^4)^9$.Since $3^4 = 81$,we have $3^{37} = 3 	ime

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(C) We need to find the remainder of $3^{37}$ when divided by $80$.We can write $3^{37} = 3 	imes (3^4)^9$.Since $3^4 = 81$,we have $3^{37} = 3 	imes (81)^9$.We can express $81$ as $(80 + 1)$.So,$3^{37} = 3 	imes (80 + 1)^9$.Using the Binomial Theorem,$(80 + 1)^9 = \binom{9}{0} 80^9 + \binom{9}{1} 80^8 + \dots + \binom{9}{8} 80 + 1$.This can be written as $80k + 1$ for some integer $k$.Therefore,$3^{37} = 3(80k + 1) = 240k + 3$.When $240k + 3$ is divided by $80$,the remainder is $3$.

The remainder when $3^{37}$ is divided by $80$ is:

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