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?
Answer with full explanation.
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Answer 1

Answer:

Option A: L = 21P

Step-by-step explanation:

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 =2×2

Prime factor of 6 =2×3

Prime factor of 8 =2×2×2

Prime factor of 10 =2×5

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=(2)⋅(2)⋅(3)⋅(2)⋅(5)=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 =(3)⋅(3)

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=(3)⋅(3)⋅(5)⋅(7)=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 =(2)3⋅(3)⋅(5)

Prime factor of Q = prime factor of 315 =(3)2⋅(5)⋅(7)

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 (L)=(2)3⋅(3)2⋅(5)⋅(7)=2520

By observation of L, we can say L=2520=(21)⋅(120)

By substituting value of P, we get

L = 21P