Question

Find the \[{\rm{LCM}}\] of 5,6 and 14.

Answer 1

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

Complete step-by-step answer:
Prime factorization: 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 \[prime \times quotient\]. Repeat the process for the quotient till you get 1 as the quotient.

Prime factorization of 5, by dividing with 5, we get:
\[5 = 5 \times 1\] …………..(1)
We should stop at this step as we got quotient 1.
Prime factorization of 6, by dividing with 2, we get:
\[6 = 2 \times 3 \times 1\] …………..(2)
We should stop at this step as we got quotient 1.
Prime factorization of 14, by dividing with 2, we get:
\[14 = 2 \times 7\]
By dividing the term 7 with 7, we can write 14 as:
\[14 = 2 \times 7 \times 1\] …………..(3)
We should stop at this step as we got quotient 1.
The above equations (1)(2)(3) implies the prime factored form of the given numbers.
Our aim is to find the least common multiple of 5,6,14.
So, by defining the least common multiple we can write it as:
Least common multiple: In arithmetic number theory the least common multiple, lowest common multiple or smallest common multiple is the smallest integer which is divisible by all the given numbers.
Process to find least common multiple: Write all the prime factored forms together. Now find the prime numbers which are repeating at least once in both or all the numbers. 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:
\[\begin{array}{l}5 = 5\\6 = 2 \times 3\\14 = 2 \times 7\end{array}\]
As the 2 is repeating in 14,6 it can be combined to one 2. Remaining all are non-repeating primes. So the least common multiple \[L.C.M.\,\,\left( {5,6,14} \right)\] can be written as:
\[LCM\,\,\left( {5,6,14} \right) = {2^1} \times {3^1} \times {5^1} \times {7^1}\]
By simplifying the above equation we get:
\[LCM = 210\]
Therefore, \[{\rm{LCM}}\] of 5,6,14 is 210

Note: Be careful while doing prime factorization, you must use only prime numbers for dividing. Here there is only one repetition so it is easy even if there are more repetitions combining all the repetitions separately, carefully as the value of least common multiple depends solely on the process of combining.