Question

Write the smallest number divisible by both \[306\] and 657.

Answer 1

Hint: The smallest number which is divisible by two or more numbers is known as LCM. It stands for Least Common Multiple. It is the method to calculate the smallest possible multiple of two or more numbers.
There are mainly two ways to calculate LCM. These are the Prime factorization method and the division method.
For the prime factorization method, we need to prime factorize the two numbers. The prime factorization of a number is the product of all such prime numbers which equals the numbers. So it is the way of finding the prime factors of the number, such that the original number is evenly divisible by these factors. Then we have to take out the common prime factors and the other uncommon prime factors.
Then LCM of the given numbers\[ = \](Product of common prime factors) \[ \times \](Product of uncommon prime factors)

Complete step-by-step answer:
Given numbers are \[306\]and \[657\]
So to get the smallest number divisible by \[306\] and \[657\], we will have to calculate the LCM of these two numbers.
Here we will calculate the LCM using prime factorization method,
For the prime factorization method , at first we have to prime factorize the two numbers which means we have to find the prime factors of that number , such that the number is evenly divisible by these factors.
So,
\[
  306 = 2 \times 3 \times 3 \times 17 \\
  657 = 3 \times 3 \times 73 \;
 \]
So here the common prime factors are \[3,3\] and the uncommon prime factors are \[2,17,73\]
We know,
 LCM of any two given numbers\[ = \](Product of common prime factors) \[ \times \](Product of uncommon prime factors)
\[ \Rightarrow \]LCM \[ = \] \[\left( {3 \times 3} \right) \times \left( {2 \times 17 \times 73} \right)\]
\[ \Rightarrow \] LCM \[ = \] \[9 \times 2482\]
\[ \Rightarrow \] LCM \[ = \]\[22,338\]
Hence the LCM of \[306\]and \[657\]is \[22,338\]
\[\therefore \]The smallest number divisible by both \[306\] and \[657\] is \[22,338\].
So, the correct answer is “22,338”.

Note: Should know about the ways of taking out the LCM. Should not commit mistakes in prime factorization of numbers. The prime numbers are those which have factors as \[1\] and the number itself whereas composite numbers are those which have factors other than \[1\] and the number itself .So be sure that we have to use only prime numbers while prime factoring the numbers, do not use composite numbers.