Question 833391
No. Not for 3 numbers
If you had just 2 numbers, each number would be the product of the greatest/highest common factor times the non-common factors found only in that number. The least common multiple would include all the factor in the greatest/highest common factor, and all the non-common factors. The product {{{HCF*LCM}}} would include the common factors twice, and the non-common factors once. So would the product of the two numbers. Therefore {{{HCF*LCM}}} would be equal to the product of the two numbers.
 
With 3 numbers, the situation is different.
Factors common to two of the numbers, but not common to all three would be included twice in the product of the three numbers, but only once in the LCM, and {{{LCM*HCF}}} .
Factors from the HCF would be included 3.
The highest common factor of 10, 12, and 24 is 2.
The least common multiple is 120.
{{{10*12*24=2880}}}
{{{10=green(2)*5}}}
{{{12=green(2)*red(6)}}}
{{{24=green(2)*red(6)*2}}}
{{{HCF=green(2)}}}
{{{LCM=green(2)*5*red(6)*2=120}}}
{{{LCM*HCF=(green(2)*5*red(6)*2)*(green(2))}}} 
{{{green(2)}}} is included twice.
The factor common to only {{{12}}} and {{{24}}} is {{{red(6)}}} and is included just once.
{{{10*12*24=(green(2)*5)*(green(2)*red(6))*(green(2)*red(6)*2)=green(2)*5*green(2)*red(6)*green(2)*red(6)*2}}}
{{{green(2)}}} is included 3 times, and {{{red(6)}}} is included twice.