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The least common multiple of two positive integers is 7!, and their greatest common divisor is 9. If one of the integers is 315, then what is the other?

MIRB16  Apr 30, 2018
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The least common multiple of two positive integers is 7!,

and their greatest common divisor is 9. If one of the integers is 315,

then what is the other?

Formula:

$$\begin{array}{|rcll|} \hline GCD(M, N) \times LCM(M, N) &=& M \times N \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline GCD(M, N) \times LCM(M, N) &=& M \times N \\ 9\times 7! &=& 315\cdot N \\\\ N &=& \dfrac{9\times 7!}{315} \\\\ &=& \dfrac{ 7!}{35} \\\\ &=& \dfrac{ 2\times 3 \times 4 \times 5 \times 6 \times 7 }{5\times 7} \\\\ &=& 2\times 3 \times 4 \times 6 \\\\ &=& 144 \\ \hline \end{array}$$

The other is 144

heureka  Apr 30, 2018