SOLUTION: Cn someone pls help me with this question? tks! Is the product of HCF n LCM of any 3 numbers always equal t the product of the 3 numbers? If yes, show 3 pairs as example.

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: Cn someone pls help me with this question? tks! Is the product of HCF n LCM of any 3 numbers always equal t the product of the 3 numbers? If yes, show 3 pairs as example.       Log On


   

Question 833391: Cn someone pls help me with this question? tks!
Is the product of HCF n LCM of any 3 numbers always equal t the product of the 3 numbers? If yes, show 3 pairs as example.
If no, show a counter-example.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
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%2ALCM would include the common factors twice, and the non-common factors once. So would the product of the two numbers. Therefore HCF%2ALCM 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%2AHCF .
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%2A12%2A24=2880
10=green%282%29%2A5
12=green%282%29%2Ared%286%29
24=green%282%29%2Ared%286%29%2A2
HCF=green%282%29
LCM=green%282%29%2A5%2Ared%286%29%2A2=120
LCM%2AHCF=%28green%282%29%2A5%2Ared%286%29%2A2%29%2A%28green%282%29%29
green%282%29 is included twice.
The factor common to only 12 and 24 is red%286%29 and is included just once.

green%282%29 is included 3 times, and red%286%29 is included twice.