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Let a, b, and c be integers that satisfy 2a+3b=52, 3b+c=41, and bc=70. Find a+b+ c.

 Jun 14, 2021
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Since $70 = 2\cdot 5\cdot 7$, there are 8 ways to factor 70 into two factors (not taking order into consideration): $1\cdot 70$, $2\cdot 35$, $5\cdot 14$, $7\cdot 10$, and four other ways similar to these but where both factors are negative.  By checking all possibilities under the constraints $3b+c=41$ and $bc=70$, we find that only $b=2$ and $c=35$ satisfy both constraints.    Substituting $b=2$ into the first constraint $2a+3b=52$ gives $a=23$.  So $a+b+c = 23+2+35 = 60$.

 Jun 15, 2021

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