Suppose ,m divided by n , then quotient q and remainder r `{:("n)m(q"),(" "-), (" "-), (" "r) , (" "):}` or ` m= nq + r , AA m,n,q, r in 1 and n ne 0 ` If a is the remainder when ` 5^(40)` us divided by 11 and b is the remainder when ` 2^(2011)` is divided by 17 , the value of a + b is

Options

A. 7
B. 8
C. 9
D. 10

Answer

c

Explanation

` 5^(40) = (5^(2))^(20) = (22 + 3)^(20) = 22 lambda + 3 ^(20), lambda in ` N
Also , ` 3^(20) = (3^(2))^(10) = (11- 2 )^(10) = 11 mu + 2^(10) , mu in ` N
Now , ` 2^(10) = 1024 = 11xx 93 + 1`
` therefore ` Remainder , a = 1
Also , ` 2^(2011) = 2^(3) (2^(4))^(502) = 2^(3) (17 -1) ^(502)`
` = 8 [(17) ^(502) - ""^(502)C_(1) (17)^(501) + ...- ""^(502)C_(501) (17) + 1]`
` = 8 (17 lambda + 1 ), lambda in N = 8 xx 17 lambda + 8 `
` therefore ` Remainder , b = 8
Hence , ` a + b = 1 + 8 = 9 `