Question

What is the least common multiple of \[25\] and \[50\]?

Answer 1

Hint: A multiple is the product of any quantity and an integer. For example, we can say that \[b\] is a multiple of \[a\] if \[b = na\], \[n\] is an integer called a multiplier. First, we will write the multiples of \[25\]and then\[50\]. Then select the common multiples and out of that we will choose the smallest multiple. There are various methods for finding the least common multiple of two given numbers. The simplest method is to list all the multiples of both the numbers and then find the smallest number common in the list of the multiples of both the numbers.

Complete answer:
Multiples of \[25\]: \[25,50,75,100,125,150,175,200......\]
These are found by multiplying \[25\] with integers such as \[1,2,3,4,5,6,7,8,9,10,11,12....\], that are called the multipliers.
Similarly, multiples of\[50\]: \[50,100,150,200,250.....\]
These are found by multiplying \[50\] with integers such as \[1,2,3,4,5,6,7,8,9,10,11,12....\]
Now we will choose the common multiples. So, the common multiples of \[25\] and \[50\]are: \[50,100....\]
Among these the smallest common multiple is: \[50\].
We can also find the smallest multiple directly by taking the LCM of \[25\] and \[50\]which is equal to: \[50\].
If we want to find more common multiples then we can find by listing more multiples of each and then taking out the common from them. But that will be a very lengthy process. In order to avoid this, we can take the first common multiple and then multiply that common multiple by integers to get a more common multiple.
For example, here the smallest common multiple is\[50\]. Now to find more common multiples we can multiply \[50\] by integers such as \[2,3,4,5,.....\]. The resulting numbers will be the common multiples of both \[25\] and \[50\].

Note:
Least common multiple (LCM) has wide ranging applications in real world as well as in mathematical questions. Knowledge of least common multiple is also used in addition and subtraction of fractions. LCM can also be calculated by using the prime factorisation method.
To find the least common multiple of 20 and 30, first we find out the prime factors of both the numbers.
Prime factors of \[25\]$ = 5 \times 5$
$ = {5^2}$
Prime factors of \[50\]$ = 5 \times 5 \times 2$
$ = 2 \times {5^2}$
Now, Least common multiple is a product of common factors with highest power and all other non-common factors. We can see that five is a common factor between both the numbers.
Hence, least common multiple of \[25\]and \[50\]$ = {5^2} \times 2$ $ = 50$