Question

Find the LCM of 336 by prime factorization method​​

Answer 1

Answer:

LCM of 336

will we

2×2×2×2×3×7

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Answer 2

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We are asked to find the L.C.M of 336 and 54

Let us use the prime factorisation method of 336 that is by using the first prime 2 then we get

$\implies$336=2×168

Now again dividing the above equation furthermore with 2 we get

$\implies$336=22×84

$\implies$336=23×42

$\implies$336=24×21

Here we can see that we cannot divide the above equation using the number 2

Now let us go for next prime number 3 then we get

$\implies$336=24×3×7

Here, we can see that the product of numbers on RHS include only prime numbers

Therefore the prime factorisation of 336 gives

$\implies$336=24×3×7

Now, let us use the prime factorisation for 54

By dividing the number 54 with first prime number 2 we get

$\implies$54=2×27

Here we can see that we cannot divide the above equation using the number 2

Now let us go for next prime number 3 then we get

$\implies$54=2×3×9

$\implies$54=2×33

Here, we can see that the product of numbers on RHS include only prime numbers

Therefore the prime factorisation of 54 gives

$\implies$54=2×33

We know that the L.C.M is given by product taking the common primes from both the numbers once and remaining time factors from both numbers. That is for example if the prime factorisation of two numbers is in the form

$\implies$a=22×3×5

$\implies$b=2×52×7

Then the L.C.M is given as

$\implies$LCM=(2×5)×(2×3×5×7)

Here the numbers in the first bracket are common primes and the numbers in the second bracket are remaining primes.

By using the above definition we get the LCM of 336 and 54 as

$\implies$LCM(336,54)=(2×3)×(23×7×32)

Now, by finding the product of above equation we get

$\implies$LCM(336,54)=3024

Therefore, the LCM of 336 and 54 is 3024.

We have a shortcut for solving this problem.

We are given that HCF of 336 and 54 is 6

we have a standard result that is

$\implies$HCF×LCM=a×b

By using the above formula we get the LCM of 336 and 54 as

$\implies$6×LCM=336×54

$\implies$LCM=181446

$\implies$LCM=3024

Therefore, the LCM of 336 and 54 is 3024.