Two Prime Roots

Two prime numbers are roots of the quadratic equation x^2-63x+k=0. What value of k makes this a true statement?
Source: mathcontest.olemiss.edu 11/13/2006

SOLUTION
Let a and b be two prime numbers that are roots of the quadratic equation x^2-63x+k=0.
We factor the equation as follows:
x^2-63x+k=\left (x-a\right )\left (x-b\right )
x^2-63x+k=x^2-bx-ax+ab 
develop the right hand side
x^2-63x+k=x^2-\left (a+b\right )x+ab  simplify

Compare the left and right hand side of the above equation
a+b=63 and ab=k

Try a few prime numbers
a=2;\;b=61  YES a solution
a=3;\;b=60  NOT a solution because is not prime
a=5;\;b=58  NO
a=7;\;b=56  NO
\cdots
Thus, k=ab=2\times 61=122

Answer: 122.

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About mvtrinh

Retired high school math teacher.
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