Question

Let ( n 2+96=m 2 ) where ( m ) Is assumed to be a positive integer.
( m 2-n 2=96 )
( Rightarrow(mathrm{m}-mathrm{n})(mathrm{m}+mathrm{n})=96=25 times 3 )
We know that, the divisors of 96 are: ( 1,2,3,4,6,8,12,16,24,32, ) 48,96
( Rightarrow ) Palrs of ( (mathrm{m}-mathrm{n}) ) and ( (mathrm{m}+mathrm{n}) ) can be 1,( 96 ; 2,48 ; 3,32 ; 4,24 ; 6,16 ; )
8,12
Subtracting first value from the second one,we get:-
( (m+n)-(m-n)=2 n )
( 2 n=95,46,29,20,10,4 )
( n=47.5,23,14.5,10,5 ) and 2
Hence, required integer values of ( n ) are 23,10,5 and ( 2 . )
So, the answer Is that ( mathrm{n} 2+96 ) is a perfect square for 4 positive Integral values.

# yul UIO 12. The number of positive integers n for which n2 + 96 is a perfect square, is (1) One (2) TWO (3) Fourts (4) Infinite

Solution