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Given that m and n are positive integers such that m is congruent to 6 (mod 9) and n is congruent to 0 (mod 9), what is the largest integer that mn is necessarily divisible by?

Jul 9, 2019

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Given that m and n are positive integers such that m is congruent to 6 (mod 9) and n is congruent to 0 (mod 9),

what is the largest integer that mn is necessarily divisible by?

I assume,

$$\begin{array}{|rclcl|} \hline m &\equiv& 6 \pmod {9} & \text{or} & m=6+9u,\quad u \in \mathbb{Z} \\ n &\equiv& 0 \pmod {9} & \text{or} & n=9v,\quad v \in \mathbb{Z} \\\\ mn &=& 9v(6+9u) \\ &=& 54v + 81uv \quad &|\quad gcd(54,81)=27 \\ \hline \end{array}$$

If m and n are positive integers, then the largest integer that mn is necessarily divisible by is 27

Jul 10, 2019