Question -
Answer -
Given:
23n┬атАУ7n тАУ 1
So, 23n┬атАУ7n тАУ 1 = 8n┬атАУ 7n тАУ 1
Now,
8n┬атАУ7n тАУ 1
8n┬а=7n + 1
= (1 + 7)┬аn
=┬аnC0┬а+┬аnC1┬а(7)1┬а+┬аnC2┬а(7)2┬а+┬аnC3┬а(7)3┬а+┬аnC4┬а(7)2┬а+┬аnC5┬а(7)1┬а+тАж +┬аnCn┬а(7)┬аn
8n┬а= 1+ 7n + 49 [nC2┬а+┬аnC3┬а(71)+┬аnC4┬а(72) + тАж +┬аnCn┬а(7)┬аn-2]
8n┬атАУ 1тАУ 7n = 49 (integer)
So now,
8n┬атАУ 1тАУ 7n is divisible by 49
Or
23n┬атАУ1 тАУ 7n is divisible by 49.
Hence proved.