If $P = 2^3 \times 3^{10} \times 5$ and $Q = 2 \times 3 \times 7$, then find he LCM of P and Q.

Given:

The given terms are $P = 2^3 \times 3^{10} \times 5$ and $Q = 2 \times 3 \times 7$.

To do :

We have to find the LCM of P and Q.

Solution :

Least Common multiple (LCM): The least common multiple of two or more numbers is the smallest non-zero common number which is a multiple of all the given numbers.

The LCM of two or more numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers.

2 occurs maximum number of times in $Q(2^5)$, 3 occurs maximum number of times in $P(3^{10})$, 5 occurs maximum number of times in $P(5^1)$ and 7 occurs maximum number of times in $Q(7^1)$

Therefore,

LCM of P and Q $=  2^5 \times 3^{10} \times 5 \times 7 = 35 \times2^5 \times 3^{10}$.

Therefore, LCM of P and Q is $35 \times2^5 \times 3^{10}$