Question

P is the LCM of 2, 4, 6, 8, 10, Q is the LCM of 1, 3, 5, 7, 9 and L is the LCM of P and Q. Then which of the following is true?
A) $\text{L=21P}$
B) $\text{L=4Q}$
C) $\text{L=63P}$
D) $\text{L=16Q}$

Answer 1

Hint: The full form of LCM is least common multiple. The name itself suggests that the number, which satisfies the condition that all the given numbers must divide this least common multiple. For finding the least common multiple the process followed is to prime factorize the given numbers, then find the number which is repeated in 2 or more numbers then round them and combine them to one number. Because we want multiples, multiplying once is enough. Find the least common multiple for given both sides of numbers then find the least common multiple for that both numbers.

Complete step-by-step answer:
Given that P is the least common multiple of 2,4,6,8,10.
To find the LCM of given numbers, we need to prime factorize all,
Prime factor of 2 $=$ 2
Prime factor of 4 $=2\times 2$
Prime factor of 6 $=2\times 3$
Prime factor of 8 $=2\times 2\times 2$
Prime factor of 10 $=2\times 5$
We can take one 2 common from all the numbers and combine it.
We can take one 2 common from 4 and 8 then combine it.
So, least common multiple P becomes as follows (after combining):
$P=\left(2\right)\cdot \left(2\right)\cdot \left(3\right)\cdot \left(2\right)\cdot \left(5\right)=120$
Given Q is least common multiple of 1,3,5,7,9
Prime factor of 1 $=$ 1
Prime factor of 3 $=$ 3
Prime factor of 5 $=$ 5
Prime factor of 7 $=$ 7
Prime factor of 9 $=\left(3\right)\cdot \left(3\right)$
We can take a 3 common from 3, 9 and combine it.
So, the least common multiple of Q of 1,3,5,7,9 after combining we get:
$Q=\left(3\right)\cdot \left(3\right)\cdot \left(5\right)\cdot \left(7\right)=315$
Now we need the least common multiple of P, Q.
Let the least common multiple of P, Q be L.
Prime factor of Q $=$ prime factor of 120 $={\left(2\right)}^3\cdot \left(3\right)\cdot \left(5\right)$
Prime factor of Q $=$ prime factor of 315 $={\left(3\right)}^2\cdot \left(5\right)\cdot \left(7\right)$
We can take one 3 common from P, Q and combine it.
We can take one 5 common from P, Q and combine it.
Least common multiple $\left(L\right)={\left(2\right)}^3\cdot {\left(3\right)}^2\cdot \left(5\right)\cdot \left(7\right)=2520$
By observation of L, we can say $L=2520=\left(21\right)\cdot \left(120\right)$
By substituting value of P, we get
$\text{L=21P}$
Therefore, option (A) is correct.

Note: Be careful while taking common. Don’t forget even if it repeats in two numbers you can take it common and combine. After getting P, Q values write then carefully. One intelligent shortcut is do not solve P, Q completely, leave it when you get them in prime factored form. You can use that directly to find the least common multiple of P, Q. After that you can write directly the value of L.C.M.