Let the sum of two positive integers be 24 . If the probability, that their product is not less than 4 3 times their greatest possible product, is n m , where gcd ( m , n ) = 1 , then n − m equals

Question

Let the sum of two positive integers be 24 . If the probability, that their product is not less than 4 3 ​ times their greatest possible product, is n m ​ , where gcd ( m , n ) = 1 , then n − m equals

TestHub · Accepted Answer

​ x + y = 24 , x , y ∈ N A M > GM ⇒ x y ≤ 144 x y ≥ 108 ​ Favorable pairs of ( x , y ) are ​ ( 13 , 11 ) , ( 12 , 12 ) , ( 14 , 10 ) , ( 15 , 9 ) , ( 16 , 8 ) , ( 17 , 7 ) , ( 18 , 6 ) , ( 6 , 18 ) , ( 7 , 17 ) , ( 8 , 16 ) , ( 9 , 15 ) , ( 10 , 14 ) , ( 11 , 13 ) ​ i.e. 13 cases Total choices for x + y = 24 is 23 ​ Probability = 23 13 ​ = n m ​ n − m = 10 ​

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