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While multiplying three real numbers, Ashok took one of the numbers as $73$ instead of $37$. As a result, the product went up by $720$. Then the minimum possible value of the sum of squares of the other two numbers is _________

Let the three numbers be $x,y$  and $z.$

Let, ashok took $z = 73$ instead of $z = 37$

So, the final product will be

$xy(73) – xy(37) = 720$

$\Rightarrow xy(73 -37) = 720$

$\Rightarrow xy(36) = 720$

$\Rightarrow \boxed {xy = 20} \quad \longrightarrow (1)$

We know that, $\text{AM}$ and $\text{GM}$ are arithmetic and geometric mean respectively between two numbers $x$ and $y$, then

$\boxed {\text{AM} \geqslant \text{GM}}$

$\Rightarrow \frac{(x+y)}{2} \geqslant \sqrt{xy}$

$\Rightarrow x+y \geqslant 2 \sqrt{xy}$

$\Rightarrow (x+y)^{2} \geqslant 4xy$

$\Rightarrow x^{2} + y^{2} + 2xy \geqslant 4xy$

$\Rightarrow x^{2} + y^{2} \geqslant 2xy$

$\Rightarrow x^{2} + y^{2} \geqslant 2(20)$

$\Rightarrow \boxed {x^{2} + y^{2} \geqslant 40}$

$\textbf{Short Method:}$

The minimum possible sum of the squares of the two numbers when $x = y.$

From equation $(1), x = y = \sqrt {20}$

so, $x^{2} + y^{2} = (\sqrt {20})^{2} + ( \sqrt {20})^{2}$

$\Rightarrow x^{2} + y^{2} = 20 + 20 = 40$

$\therefore$ The minimum possible value of the sum of squares of the other two numbers is $40.$

Correct Answer $:40$

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