Hi All,
This question is interesting in that it combines Algebra rules with Number Property rules. While you don't have to do much advanced math to answer it, you DO need to recognize the patterns that the equation is built on.
We're given Y^2 -63Y + X = 0
At first glance, this looks a bit "scary", but it should remind you of a Quadratic.
We're told that there are 2 roots and that they are BOTH PRIMES. This severely limits the possibilities....
Y^2 - 63Y + X = 0
Since the "middle term" is -63Y, the two parentheses are either BOTH negative or 1 negative/1 positive...
(Y - ?)(Y - ?) = 0
or
(Y - ?)(Y + ?) = 0
-63Y is the SUM of the roots. BOTH roots are PRIME though, so we have to think about what happens when you add/subtract odds and evens
EVEN + EVEN = EVEN
EVEN - EVEN = EVEN
ODD + ODD = EVEN
ODD - ODD = EVEN
None of these outcomes matches -63, which is an ODD number. This means that the two primes MUST be 1 odd and 1 even. The ONLY even prime is 2....so now the possibilities are...
(Y - ?)(Y - 2) = 0
or
(Y - ?)(Y + 2) = 0
To end up with -63Y as the middle term, the missing value in each of the above 2 examples would have to be.... -61 or +65....but 65 is NOT prime, so the only option is -61....
(Y - 61)(Y - 2) = 0
FOILing this out, we get...
Y^2 - 63Y + 122 = 0
X can ONLY be +122, so there's only one answer to the question.
Final Answer:
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Rich
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