7⁷¹ = (7⁴)¹⁷ × 7³
Hence, unit digit in 7⁷¹
= Unit digit in [(7⁴)¹⁷ × 7³]
= Unit digit in [(7 × 7 × 7 × 7 )¹⁷ × 7³]
= Unit digit in [(9 × 9)¹⁷ × 7³] (∵ 7 × 7 = 49 and 9 is the unit digit of 49)
= Unit digit in [1¹⁷ × 7³] (∵ 9 × 9 = 81 and 1 is the unit digit of 81)
= Unit digit in [1 × 7³] (∵ 1¹⁷ = 1)
= Unit digit in [7³]
= Unit digit in [7 × 7 × 7]
= Unit digit in [9 × 7](∵ 7 × 7 = 49 and 9 is the unit digit of 49)
= 3 (∵ 9 × 7 = 63 and 3 is the unit digit of 63)