JMO

What is the Remainder in  7⁵ ÷ 6

There are two ways of solving this question:

first method:

7⁵ - 1⁵ is always divisible by 7 – 1 = 6

So, 7⁵ - 1⁵ = 6Q

Or, 7⁵ = 6Q + 1

So, Remainder = 1

 

second method:

(1 + 6)⁵ = 1⁵ + 5.1⁴.6….. + 5.1.6⁴ +6⁵

Since all terms have 6, except the first term, we have

(1 + 6)⁵ ÷ 6 = (1⁵ + 5.1⁴.6….. + 5.1.6⁴ +6⁵) ÷ 6 = Q + 1⁵ ÷ 6 = Q + 1 ÷ 6  ( use pascals triangle to write the coefficients)

Remainder = 1