(1) can anyone solve my question.

 

 

level 3rd. 

there IS a mistake in the question! it is not divisible by 73. (arihant IS known to be wrong at times.)

 

maybe the problem is to prove that is divisible by 73?

I said earlier, there is a mistake in the prob bcos of this:

 

72!/36! - 1 = (37*38*39*...*72) - 1 = (73-36)*(73-35)*...*(73-1) = 73*k + 36! - 1.

 

It remains to prove that 36! - 1 is divisible by 73. This cannot be true from a property arising from Wilson's Theorem (Number Theory)  that if p is a prime then (p-1/2)! ²+1 is divisible by p if p is of the form 4k+1. Now 73 is such a prime, which gives (36!)²+1 is divisible by 73.

 

Now, if 36!-1 is div by 73 we must have from the above that 2.36! and hence 36!is also divisible by 73 i.e. 36!-1 and 36! are both div by 73 which is impossible.

 

ya exactly...

 

.

Notice also

 

it is quite clear, 73 doesnt figure in it...