Question

Find the remainder when 16^53 is divided by 7​

Answer 1

we have to find the remainder when 16⁵³ is divided by 7.

solution : 16⁵³ can be written as (2 × 7 + 2)⁵³

now (16)⁵³/7 = (2 × 7 + 2)⁵³/7

remainder of 16⁵³/7 = remainder of (2 × 7 + 2)⁵³/7 = remainder of 2⁵³/7

[ because 2 × 7 is divisible by 7 so remainder is zero ]

now 2⁵³ = 2².2⁵¹ = 4.(2³)^17 = 4.(8)^17

now 2⁵³/7 = 4.(8)^17/7 = 4.(1 × 7 + 1)^17/7

remainder of 2⁵³/7 = remainder of 4.(1 × 7 + 1)^17/7 = remainder of 4.(1)^17/7 = remainder of 4/7 = 4

Therefore the remainder of 16⁵³/7 is 4.