If `7^(103)` is divided by 25 , find the remainder .

Explanation

We have , ` 7^(103) = 7*7^(102) = 7*(7^(2)) ^(51) = 7 (49)^(51) =7 (50-1)^(51)`
`= 7 [(50)^(51) - ""^(51)C_(1)(50)^(50) + ""^(51)C_(2) (50)^(49) - ... -1] `
`= 7 [(50)^(51) - ""^(51)C_(1)(50)^(50) + ""^(51)C_(2) (50)^(49) - ...+ ""^(51)C_(50)(50)]- 7 - 18 + 18`
`= 7 [50((50)^(50) - ""^(51)C_(1)(50)^(49) + ""^(51)C_(2) (50)^(48) - ...+ ""^(51)C_(50))]- 25 + 18 `
` = 7 [50k] - 25 + 18 ` , where k is an integer .
` = 25[ 14k - 1] + 18 = 25 p + 18 " "` [ where p is an integer ]
Now , ` (7^(103))/(25) = p + (18)/(25) ` . Hence , the remiinder is 18 .