If a and b are the lengths of two sides of a triangle such that the product ab = 24, where a and are integers, then how many such triangles are possible?

15

Let us consider that 3rd side of triangle is = x

abx is a triangle iff ( a - b ) < x < ( a + b )

It is given that :- ab = 24

So, Possible values of a and b are :-

( 1 , 24 )

( 24 - 1 ) < x < ( 24 + 1 ) = 23 < x < 25 so, x = 24 

( 2 , 12 )

( 12 - 2 ) < x < (12 + 2 ) = 10 < x < 14  so, x = 11, 12, 13

( 3, 8 )

( 8 - 3) < x < ( 8 + 3) = 5 < x < 11  so, x = 6 , 7, 8, 9, 10

( 4 , 6 )

( 6 - 4) < x < ( 6 + 4) = 2 < x < 10  so, x = 3 , 4, 5, 6, 7, 8, 9

So, Total number of values of x = 16

So, Possible number of triangles is = 16