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Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write q p as l+ m n , where l, m, n are positive integers, m < n and GCD (m, n)=1 . Find the maximum value of l+m+n .

Correct answer

32

Step-by-step solution

aligned & q^2+3 p=197 p^2+q & 197 p^2=q(q-1)+3 p aligned So, q-1= p array ll & 197 p^2=( p+1) p+3 p & 197 p= ^2 p+ +3 & p= +3 197- ^2 & =14, p=17 & q=17 14+1=239 array So, q p =14+ 1 17 I+m+n=32

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