Suppose ,m divided by n , then quotient q and remainder r
or ` m= nq + r , AA m,n,q, r in 1 and n ne 0 `
If `13^(99)` is divided by 81 , the remainder is

Options

A. (a)13
B. (b)23
C. (c)39
D. (d)55

Answer

d

Explanation

` 19^(93) - 13^(99) ` = (odd number ) - (odd number ) = even number
` therefore 19^(93) - 13^(99) ` is divisible by 2.
Now , ` 19^(93) - 13^(99) ` is divisible by 2 .
Now , ` 19^(93) - 13^(99) = (18 +1) ^(93) - (12 + 1)^(99) `
`=[(18) ^(93) + ""^(93)C_(1) (18) ^(92) + ""^(93)C_(2) (18)^(91) ...+ ...+ ""^(93)C_(92) (18) + 1]`
` - [(12)^(99) + ""^(99)C_(1) (12)^(98) + ""^(99)C_(2) (12)^(97) + ...+ ""^(99)C_(98)(12) + 1]`
` = (18)^(2) lambda + ""^(93)C_(1)xx 18 - (12)^(2) mu - ""^(99)C_(1) (12) `
When ` lambda " and " mu ` are integers
` = (18)^(2) lambda - (12)^(2) mu + 486`
` 81xx 4 lambda - 12^(2) (""^(99)C_(2) + 12. ""^(99)C_(3)) + 81 p + 486`
` = 81 ` (integer), where p is an integer
But 2 and 91 are co-prime .
` therefore 19^(93) - 13^(99) ` is divisible by 162 .