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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Suppose an integer r , a natural number n and a prime number p...

Suppose an integer r , a natural number n and a prime number p satisfy the equation

7 x^2-44 x+12=p^n

. Find the largest value of

p

.

Correct answer

47

Step-by-step solution

\begin{array}{ll}7 x^2-44 x+12=p^n & \\ 7 x^2-42 x-2 x+12=p^n & \\ (7 x-2)(x-6)=p^n & \\ 7 x-2=p^\alpha \text { and } x-6=p^\beta & \\ (7 x-2)-7(x-6)=p^\alpha-7 p^\beta & \\ 40=p^\alpha-7 p^\beta & \\ \text { If } \alpha, \beta \in N, p \text { is divisors of } 40 & \Rightarrow p=2 \text { or } 5 \\ \text { If } p=2,40=2^\alpha-7.2^\beta & \Rightarrow 2^3 .5=2^\alpha-7.2^\beta \\ \Rightarrow \beta=3 \text { and } 2^\alpha=40+56 & \Rightarrow \alpha \notin Z \text { hence not possible } \\ \text { If } p=5 \text { then } 40=5^\alpha-7.5^\beta & \Rightarrow 2^3 .5=5^\alpha-7.5^\beta \\ \Rightarrow \beta=1 \text { and } 5^\alpha=40+35 & \Rightarrow \alpha \notin Z \text { hence not possible } \\ \text { so } \beta=0 \Rightarrow p^\alpha=47 & \Rightarrow p=47 \text { and } \alpha=1\end{array}

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