The least common multiple of a number "n" and 6 is 24. What is the value for "n"?

Thursday, April 9, 2015

The least common multiple of a number "n" and 6 is 24. What is the value for "n"?

Let's consider the prime factorization of both given numbers, $\displaystyle{6}$ and $\displaystyle{24}.$

It is clear that $\displaystyle{6}={{2}}^{{1}}\cdot{{3}}^{{1}}$ and $\displaystyle{24}={{2}}^{{3}}\cdot{{3}}^{{1}}.$

Hence the number $\displaystyle{n}$ must contain $\displaystyle{2}$ exactly in degree $\displaystyle{3}$ in its prime factorization. If it would have $\displaystyle{2}$ in greater degree, the LCM of $\displaystyle{6}$ and $\displaystyle{n}$ would have $\displaystyle{2}$ in that greater degree, and if in less, then in less.

Also $\displaystyle{n}$ may contain $\displaystyle{3}$ in degree not greater than $\displaystyle{1}.$ It may contain $\displaystyle{3}$ in degrees $\displaystyle{0}$ or $\displaystyle{1},$ because $\displaystyle{6}$ already have $\displaystyle{{3}}^{{1}}$ and $\displaystyle{24}$ also.