QUESTION

Find the LCM of the numbers 50 and 60.

Hint: We know that LCM stands for least common multiple. Multiple of a number x means that the multiple is a product of a natural number and x itself. If we take two numbers a and b, LCM of a and b means the smallest possible number which is multiple of both a and b.

Let us proceed to find the LCM of 50 and 60. A naive approach is to list down some multiples of both numbers and find out the lowest common of them. This method may not work always, but is the simplest method of all.
Let us first find out the multiples of both 50 and 60.
Multiples of 50 - 50,100,150,200,250,300,350,400,450………….and so on.
Multiples of 60 - 60,120,180,240,300,360,420,480,540………….and so on.
We can easily see that 300 is the smallest value which is common in both .
So, we can conclude here that 300 is the LCM of 50 and 60.
But we can clearly see that this method will be a bit lengthy if we don’t find the common multiple in the first few multiples.
Another approach is by using prime factorization of two numbers.
$50=2\times 5\times 5$
$60=2\times 2\times 3\times 5$
So, what we do is just multiply each factor the greatest number of times it appears in either number.
Now, let us see for the number 2 , it appears one time in 50 and two times in 60. So, our answer includes 2 times 2.Again seeing for the number 3, it appears zero times in 50 and one time in 60.So, our answer includes one time 3. Again seeing for the number 5, it appears two times in 50 and one time in 60. So, our answer includes 5 two times. So, we now combine all the things which we have included in our answer.
$2\times 2\times 3\times 5\times 5=300$