n being any odd number greater than 1, n65 - n is always divisible by :
A. 5
B. 13
C. 24
D. None of these
Answer: Option C
Solution(By Examveda Team)
$$\eqalign{ & \Leftrightarrow {n^{65}} - n \cr & = n\left( {{n^{64}} - 1} \right) \cr & = n\left( {{n^{32}} - 1} \right)\left( {{n^{32}} + 1} \right) \cr & = n\left( {{n^{16}} - 1} \right)\left( {{n^{16}} + 1} \right)\left( {{n^{32}} + 1} \right) \cr} $$$$ = n\left( {{n^8} - 1} \right)\left( {{n^8} + 1} \right)\left( {{n^{16}} + 1} \right)$$ $$\left( {{n^{32}} + 1} \right)$$
$$ = n\left( {{n^4} - 1} \right)\left( {{n^4} + 1} \right)\left( {{n^8} + 1} \right)$$ $$\left( {{n^{16}} + 1} \right)$$ $$\left( {{n^{32}} + 1} \right)$$
$$ = n\left( {{n^2} - 1} \right)\left( {{n^2} + 1} \right)\left( {{n^4} + 1} \right)$$ $$\left( {{n^8} + 1} \right)$$ $$\left( {{n^{16}} + 1} \right)$$ $$\left( {{n^{32}} + 1} \right)$$
$$ = \left( {n - 1} \right)n\left( {n + 1} \right)\left( {{n^2} + 1} \right)\left( {{n^4} + 1} \right)$$ $$\left( {{n^8} + 1} \right)$$ $$\left( {{n^{16}} + 1} \right)$$ $$\left( {{n^{64}} + 1} \right)$$ $$\left( {{n^{32}} + 1} \right)$$
Clearly, (n - 1), n and (n + 1) are three consecutive numbers and they have to be multiples of 2, 3 and 4 as n is odd.
Thus, the given number is definitely a multiple of 24
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Related Questions on Number System
Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:
A. 12, 24, 36
B. 11, 22, 33
C. 12, 24, 32
D. 5, 10, 15
यदि n postive even number है तो यह किससे विभाज्य होगा?