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sqrt(79 + 24*sqrt(7)) can be written in the form a + b*sqrt(c), where a, b and c are integers and c has no factors which is a perfect square of any positive integer other than 1. Find abc.

 May 10, 2021
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Let $b\sqrt{c}=\sqrt{d}$

 

So $(a+\sqrt{d})^2=79+24\sqrt{7}\implies a^2+d+2a\sqrt{d}=79+24\sqrt{7}$. Let's try $d=7$, then $a^2+7=79, 2a=24$, which doesn't work. Now trying $\sqrt{d}=3\sqrt{7}$, then $d=63$ and $a^2=16, a=4$, which indeed works. Then $a+\sqrt{d}=4+\sqrt{63}=4+3\sqrt{7}$ and $abc=12*7=\boxed{84}$.

 May 10, 2021

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