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If 9^7-7^9 is divisible by 2^n , then fi...

If `9^7-7^9` is divisible by `2^n ,` then find the greatest value of `n ,w h e r en in Ndot`

Text Solution

Verified by Experts

We have
`9^(7)-7^(9) = (1+8)^(7) - (1-8)^(9)`
`= (1+.^(7)C_(1)8^(1)+.^(7)C_(2)8^(2)+"….."+.^(7)C_(7)8^(7)) - (1-.^(9)C_(1)8^(1)+.^(9)C_(2)8^(2)-"……."-.^(9)C_(9)8^(9))`
`= 16xx8+64[(.^(7)C_(2)+"….."+.^(7)C_(7)8^(5))-(.^(9)C_(2)-"……"-.^(9)C_(9)8^(7))]`
`= 64k` (where k is some integer )
Therefore, `9^(7) - 7^(9)` is divisible by `64`.
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