Question

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

JEE/Engineering Exams

Maths

Solution

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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.