Discussion

Explanation:

Let us find the pattern that remainders follow when successive powers of 3 are divided by 7.

Remainder when 7¹/10 = 7.
Remainder when 7²/10 = 9.
Remainder when 7³/10 = 3.
Remainder when 7⁴/10 = 1.
Remainder when 7⁵/10 = 7.
Remainder when 7⁶/10 = 9.

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

$R\left[\frac{7^{83}}{10}\right]$ = $R\left[\frac{7^{80}\times 7^3}{10}\right]$ = $R\left[\frac{7^{80}}{10}\right]$ × $R\left[\frac{7^3}{10}\right]$ = 1 × 3 = 3.

Alternately,

Remainder of any number when divided by 10 is same as the last digit of that number.

∴ We have to find the last digit of 7⁸³.

Last of 7⁸³ = last digit of  7⁸⁰ × 7³ (Cyclicity of last digit of powers of 7 is 4.)

= last digit of 7³

= 3

Hence, 3.

» Your doubt will be displayed only after approval.