Prove that if ' p ' is a prime number greater than 2, then 
− 2 p+1 is divisible by p, where [ . ] denotes greatest integer function
Text Solution
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− 2 p+1
Let 
so [ 
] = I , where I is an integer and f ∈ (0, 1)
∈ (0, 1)
2[ p C 0 2 p + p C 2 2 p–2 
+ ......] = I + f – f '
⇒ f ' – f = 0 ⇒ f = f ' ⇒ 
– 2 p+1 = 2[ p C 0 2 p + p C 2 2 p–2 .5 + ........] –2 p+1
= p C 2 .2 p–1 .5 + p C 4 2 p–3 .5 2 + ........
This is always divisible by p because for a prime number p, p C r (1 < r < p) is always divisible by p.
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