Discussion

Explanation:

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

Remainder of 3¹/7 = 3.
Remainder of 3²/7 = 2.
Remainder of 3³/7 = 6.
Remainder of 3⁴/7 = 4.
Remainder of 3⁵/7 = 5.
Remainder of 3⁶/7 = 1.
Remainder of 3⁷/7 = 3.
Remainder of 3⁸/7 = 2.

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

∴ $R\left[\frac{3^{100}}{7}\right]$ =  $R\left[\frac{3^{96}\times 3^4}{7}\right]$ = $R\left[\frac{3^{96}}{7}\right]\times R\left[\frac{3^4}{7}\right]$ = 1 × 4 = 4.

⇒ The remainder is 4.

Alternately,

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

Remainder of 3¹/7 = 3.
Remainder of 3²/7 = 2.
Remainder of 3³/7 = 6 = -1. (Concept of negative remainder)

∴ R(3³/7) = -1

∴ $R\left[\frac{3^{100}}{7}\right]$=  $R\left[\frac{3^{99}\times 3^1}{7}\right]$ = $R\left[\frac{3^{99}}{7}\right]\times R\left[\frac{3^1}{7}\right]$ = ${\left(R\left[\frac{3^3}{7}\right]\right)}^{33}$ × 3 = (-1)³³ × 3 = -3 ≡ 7 - 3 = 4.

Hence, 4.

» Your doubt will be displayed only after approval.