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

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write...

Suppose the prime numbers

p

and

q

satisfy

q^2+3 p=197 p^2+q

. Write

\frac{q}{p}

l+\frac{m}{n}

, where

l, m, n

are positive integers,

m < n

and

\operatorname{GCD}(m, n)=1

. Find the maximum value of

l+m+n

Correct answer

Step-by-step solution

\begin{aligned} & q^2+3 p=197 p^2+q \\ & \Rightarrow \quad 197 p^2=q(q-1)+3 p \end{aligned}

So,

q-1=\lambda p

\begin{array}{ll} \therefore & 197 p^2=(\lambda \cdot p+1) \lambda p+3 p \\ \Rightarrow & 197 p=\lambda^2 p+\lambda+3 \\ \therefore & p=\frac{\lambda+3}{197-\lambda^2} \\ \therefore & \lambda=14, p=17 \\ \Rightarrow & q=17 \times 14+1=239 \end{array}

So,

\quad \frac{q}{p}=14+\frac{1}{17}

\therefore \quad I+m+n=32

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