question_answer

If two positive integers x and y are expressible in terms of primes as \[x={{p}^{2}}{{q}^{3}}\] and \[y={{p}^{3}}q\], what can you say about their LCM and HCF. Is LCM a multiple of HCF? Explain.

Answer:

Given,                \[x={{p}^{2}}{{q}^{3}}\]

\[=p\times p\times q\times q\times q\]

And                  \[y={{p}^{3}}q\]

\[=p\times p\times p\times q\]

\[\therefore HCF=p\times p\times q={{p}^{2}}q\]

And                  \[LCM=p\times p\times p\times q\times q\times q={{p}^{3}}{{q}^{3}}\]

\[\Rightarrow LCM=p{{q}^{2}}(HCF)\]

Yes, LCM is a multiple of HCF.

Explanation:

Let                    \[a=12={{2}^{2}}\times 3\]

\[b=18=2\times {{3}^{2}}\]

\[\therefore HCF=2\times 3=6\]                ?(i)

\[LCM={{2}^{2}}\times {{3}^{2}}=36\]

\[LCM=6\times 6\]

\[LCM=6\text{ (}HCF)\]                         [From (i)]

Here LCM is 6 times HCF.