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(C) Given the equation: $6^m + 2^{m+n} \cdot 3^w + 2^n = 332$.We can rewrite the equation as $6^m + 2^m \cdot 2^n \cdot 3^w + 2^n = 332$,which simplif

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$19$

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(C) Given the equation: $6^m + 2^{m+n} \cdot 3^w + 2^n = 332$.We can rewrite the equation as $6^m + 2^m \cdot 2^n \cdot 3^w + 2^n = 332$,which simplifies to $6^m + 2^n(2^m \cdot 3^w + 1) = 332$.If $m=2$,then $6^2 + 2^n(2^2 \cdot 3^w + 1) = 332$.$36 + 2^n(4 \cdot 3^w + 1) = 332$.$2^n(4 \cdot 3^w + 1) = 296$.Since $296 = 8 	imes 37 = 2^3 	imes 37$,we have $2^n = 2^3$,so $n=3$.Also,$4 \cdot 3^w + 1 = 37$,which implies $4 \cdot 3^w = 36$,so $3^w = 9$,which gives $w=2$.Since $m=2, n=3$ are positive integers,the values satisfy the equation.Now,calculate $m^2 + mn + n^2 = (2)^2 + (2)(3) + (3)^2 = 4 + 6 + 9 = 19$.

Suppose $m, n$ are positive integers such that $6^m + 2^{m+n} \cdot 3^w + 2^n = 332$. The value of the expression $m^2 + mn + n^2$ is

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