Divisibility

Probability Level 4

Let n n be a positive integer and k k be a positive integer larger than 1. If 3 2 n + 2 − 8 n − 9 3^{2n+2} - 8n-9 is divisible by k k , find the total number of possible values of k k .

Check out the set Binomial Thm. !
3 2 None of these choices 6 7 4

Sabhrant Sachan by Sabhrant Sachan , 21, India

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1 solution

Sabhrant Sachan
May 16, 2016

3 2 n + 2 − 8 n − 9 3^{2n+2}-8n-9 can be written as ( 1 + 8 ) n + 1 − 8 n − 9 (1+8)^{n+1}-8n-9

⟹ [ 1 + ( n + 1 1 ) 8 + ( n + 1 2 ) 8 2 + ⋯ + ( n + 1 n ) 8 n + 8 n + 1 ] − 8 n − 9 ⟹ 1 + 8 n + 8 + [ ( n + 1 2 ) 8 2 + ⋯ + ( n + 1 n ) 8 n + 8 n + 1 ] − 8 n − 9 ⟹ ( n + 1 2 ) 8 2 + ⋯ + ( n + 1 n ) 8 n + 8 n + 1 ⟹ 64 [ ( n + 1 2 ) + ⋯ + ( n + 1 n ) 8 n − 2 + 8 n − 1 ] \implies \left[1+\dbinom{n+1}{1}8+\dbinom{n+1}{2}8^2+\cdots+\dbinom{n+1}{n}8^n+8^{n+1}\right]-8n-9\\ \implies 1+8n+8+\left[\dbinom{n+1}{2}8^2+\cdots+\dbinom{n+1}{n}8^n+8^{n+1}\right]-8n-9\\ \implies \dbinom{n+1}{2}8^2+\cdots+\dbinom{n+1}{n}8^n+8^{n+1} \\ \implies 64[\dbinom{n+1}{2}+\cdots+\dbinom{n+1}{n}8^{n-2}+8^{n-1}]

The given Expression is divisible by 64 , Possible values of k k are 2 , 4 , 8 , 16 , 32 , 64 2,4,8,16,32,64 . Our answer is 6 \boxed{6}

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