What is the LCM of 4 and 6 and 14?

Answer 1

Prime factorization of 4 = 2 x 2 = 22

Prime factorization of 6 = 2 x 3 = 21 x 31

Prime factorization of 14 = 2 x 7 = 21 x 71

Now we need to look for the highest power of each prime factor:

The highest power of prime factor 2 is 2.

The highest power of prime factor 3 is 1 and that of 7 is also 1.

And LCM is equal to product of prime factors with highest power.

So, LCM = 22 x 31 x 71 = 4 x 3 x 7 = 84.

But why we choose highest power?

If we multiply 4, 6 and 14 then we get a common multiple of 4, 6 and 14.

Any number is a multiple of 4, 6 and 14 if in its prime factorization we have 2 x 2, 2 x 3 and 2 x 7.

Since 4 has more 2s so 6 and 14 can share 2s of 4. And we can remove the 2s of 6 and 14 and we still get the common multiple.

i.e. (2 x 2) x (2 x 3) x (2 x 7) = 2 x 2 x 3 x 7.

Let us show that 2 x 2 x 3 x 7 has 2 x 2, 2 x 3 and 2 x 7.

i) (2 x 2) x 3 x 7

ii) 2 x (2 x 3) x 7

iii) 2 x 3 x (2 x 7)

Now we can't remove any more prime factors so we have the least common multiple.

So we have to look for the prime factors which can be shared in the factorization of multiple numbers.

In this case 6 and 14 share 2s of 4.

Since 4 has more more 2s so we choose 2s of 4 for the purpose for sharing.

Or we can say we choose 22. Now we can see the reason behind choosing the highest power.

Answer 2

84