MHB What Is the Correct Method to Find the LCM of 12 and 56?

[Casio1]

I did actually think I had this off to a T sort of speak, but seems I have run into a problem and am now unsure?

Find the LCM of 12 and 56.

This is how I do them but if incorrect I would appreciate the correct notation being pointed out to me;)

12/2 = 6 My prime factors of 12 are therefore 2, 6

56/2 = 28, and 28/2 = 14, and 14/2 = 7

my prime factors are; 2, 6, 7, 14

My common factors being 2

my lowest common multiple being = 1176?

12/4 = 3, therefore 3,4

56/14 = 4 therefore 4,14

common factors are 4

Lowest common multiple is 3 x 4 x 14 = 168
:D
so I am assuming there is a specific notation (method) to finding the LCM's without mistakes?

 

[Reckoner]

my prime factors are; 2, 6, 7, 14

6 and 14 cannot be prime factors, for they are not prime!

\(12 = 2\cdot6 = 2\cdot2\cdot3 = 2^2\cdot3\)

\(56 = 2\cdot28 = 2\cdot2\cdot14=2\cdot2\cdot2\cdot7=2^3\cdot7\)

Now we take the highest power of each factor. Our least common multiple is therefore \(\mathrm{LCM}(12,56) = 2^3\cdot3\cdot7 = 168\).

 

[Casio1]

6 and 14 cannot be prime factors, for they are not prime!

\(12 = 2\cdot6 = 2\cdot2\cdot3 = 2^2\cdot3\)

\(56 = 2\cdot28 = 2\cdot2\cdot14=2\cdot2\cdot2\cdot7=2^3\cdot7\)

Now we take the highest power of each factor. Our least common multiple is therefore \(\mathrm{LCM}(12,56) = 2^3\cdot3\cdot7 = 168\).

Thanks for point that out to me I completely missed that point, i.e. 14 is not a prime.

So should I take it then that the correct way to prime factorise intergers is to divide them always by prime numbers?

90 / 2 = 45, then 45 / 3 = 15, then 15 / 5 = 3

 

[Reckoner]

So should I take it then that the correct way to prime factorise intergers is to divide them always by prime numbers?

90 / 2 = 45, then 45 / 3 = 15, then 15 / 5 = 3

You can factor the integers however you like. The key is that you don't stop factoring until all the factors are prime. When you get to that point, you will have found the prime factorization of the integer, which is what we want.

So we could factor 90 as \(90 = 9\cdot10\), but we have to keep going, because 9 and 10 are not prime:

\(90 = 9\cdot10 = (3\cdot3)\cdot(2\cdot5) = 2\cdot3^2\cdot5\).

We stop here because 2, 3, and 5 are all prime.Of course, there are other methods for calculating the least common multiple that do not require factorization. If you only have two numbers, one way to find their LCM is to divide the product of the numbers by their greatest common divisor:

\[\mathrm{LCM}(a, b) = \frac{ab}{\mathrm{GCD}(a, b)}.\]

Another quick algorithm that's easy to do mentally is to take successive multiples of the bigger number until you find one that is divisible by the other number(s). For example, to find the LCM of 12 and 56, we would take multiples of 56:

\(56\cdot1 = 56\), which is not divisible by 12
\(56\cdot2 = 112\), which is not divisible by 12
\(56\cdot3 = 168\), which is divisible by 12. 168 is our least common multiple.