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If 6^(83) + 8^(83) is divided by 49 , t...

If ` 6^(83) + 8^(83)` is divided by 49 , the raminder is

A

0

B

14

C

35

D

42

Text Solution

Verified by Experts

The correct Answer is:
c
`6^(83) + 8^(83) = (7-1)^(83) + (7+1)^(83) `
`= 2(7^(83) + ""^(83)C_(2) *7^(81) + ""^(83)C_(4) *7^(79) + ...+ ""^(83)C_(80)7^(3) + ""^(83)C_(82)7)`
` = 2 { 49 m + ""^(83)C_(82) *7}`
where , m is an integer
`= 98m + 2*""^(83)C_(1) *7 = 98m + 2*83 *7`
`= 98m + 2 (77 + 6)*7 = 49 (2m + 22) + 84 `
`= 49 (2m + 22) + 49 + 35 `
` = 49 (2m+ 23) + 35 = 49 n + 35 `
where n is an integer .
Hence , the remainder is 35 .
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