Question

Find the LCM of 2, 4 and 8.

Answer 1

Hint: First by using the definition of prime factorization, find the prime factorized value for all the 3 given numbers. Then use the definition of least common multiple to find the \[\text{LCM}\] of all these numbers \[\text{LCM}\] is the required result.

Complete step-by-step answer:
 In number theory, prime factorization is the decomposition of a composite number into a product of few prime numbers which are smaller than the original number. This process is carried out by dividing with prime numbers and finding quotients. Thus, writing numbers as \[\text{prime }\times \text{ quotient}\]. Repeat the process for the quotient till you get 1 as the quotient.
Prime factorization of 2, by dividing with 2, we get it as:
\[2=2\times 1.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }\left( \text{1} \right)\]
We have to stop at this step as we got 1 as quotient.
Prime factorization of 4, by dividing with 2, we get it as:
\[4=2\times 2\]
By dividing term 2 with 2, we can write 4 as:
\[4=2\times 2\times 1.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }\left( \text{2} \right)\]
We have to stop at this step as we got 1 as quotient.
Prime factorization of 8, by dividing with 2, we get it as: \[8=4\times 2\]
By dividing term 4 with 2, we can write 8 as: \[8=2\times 2\times 2\]
By dividing term 2 with 2, we can write 8 as:
\[8=2\times 2\times 2\times 1.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }\left( \text{3} \right)\]
We have to stop here as we got 1 as the quotient.
Least common multiple: - In arithmetic, number theory the least common multiple, lowest common multiple or smallest common multiple is the smallest integer k which is divisible by all the given numbers.
Process to find least common multiple: - Write all the prime factorized forms together. Now find the prime number which is repeating at least once in both or all the numbers. Just combine all the repeating primes into one prime. This way you get the least multiple in terms of prime. Just simplify it to get the least common multiple.
By writing all the 3 equations we found together, we get:
\[\begin{align}
  & 2=2 \\
 & 4=2\times 2 \\
 & 8=2\times 2\times 2 \\
\end{align}\]
So, we can take one 2 as repeating in all 3 and combine we can take another 2 as repeatedly in 4, 8 and combine. After these 2 steps of combining write \[\text{LCM}\] (2, 4, 8) as: \[LCM={{2}^{1}}\times {{2}^{1}}\times {{2}^{1}}=8\].
Therefore, the least common multiple of 2, 4, 8 is 8.

Note: Be careful with the method of prime factorization, you must only use prime numbers for dividing. Here there are 2 repetitions so find them properly and then combine them. As the whole process of prime factorization depends on the step of combining we must do it carefully. We can also find the LCM using the division method, where all the three numbers are divided with prime factors till the remainder for each number is 1 or a prime number. Then multiplying all the factors which act as divisors, we will get the LCM.