Discussion

Explanation:

Let us find the pattern that remainders follow when successive powers of 7 are divided by 8.

Remainder of 7¹/8 = 7.
Remainder of 7²/8 = 1.
Remainder of 7³/8 = 7.
Remainder of 7⁴/8 = 1.

∴ We find that the remainders are repeated after every two powers. 

∴ Remainder of 7⁶⁴ when divided by 8 is the same as 7² when divided by 8, since 64 is a multiple of 2. 

⇒ The remainder is 1.

Alternately,

$R\left[\frac{7^{64}}{8}\right]$ = ${\left[R\left(\frac{7}{8}\right)\right]}^{64}$ = [-1]⁶⁴ = 1

Hence, 1.

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