Question

The least common multiple of two numbers is 60 and one of the numbers is 7 less than the other. What are the numbers?

Answer 1

Hint:We will use the concepts of LCM (Least Common Multiple) to solve this problem. A common multiple for two or more numbers which is least is known as least common multiple. It is also called the least common divisor. We will learn about the method of finding LCM of numbers.

Complete step by step solution:
In mathematics, the LCM of two numbers is defined as the least number divisible by the both given numbers. For example, take numbers 4 and 6.
Let us find the LCM of these two numbers.
Firstly, multiple of 4 are \[4,8,12,16,20,24,28,......\]
Multiples of 6 are \[6,12,18,24,30,......\]
Here, we can observe that the least multiple common for both 4 and 6 is \[12\] .
So, LCM of 4 and 6 is \[12\]
\[ \Rightarrow LCM(4,6) = 12\]
Now, in the question, it is given as LCM of two numbers is 60.
So, let the two numbers be \[a{\text{ and }}b\]
\[ \Rightarrow LCM(a,b) = 60\]
From this, we can conclude that 60 is divisible by both \[a{\text{ and }}b\] .
And the divisors of \[60\] are \[1,2,3,4,5,6,10,12,15,20,30,60\]
So, the values of \[a{\text{ and }}b\] are among these divisors.
In the question, it is also given as one of the numbers is 7 less than the other.
So, the difference of \[a{\text{ and }}b\] is 7.
So, we can make a guess that the two numbers are \[{\text{3 and 10}}\] or \[{\text{5 and 12}}\] .
But, the LCM of \[{\text{3 and 10}}\] is 30.
So, the numbers are \[{\text{5 and 12}}\] .

Note:
In finding LCM, we get many common multiples, but we need to consider only the least once.
For example, \[{\text{3 and 10}}\] have 30 as well as 60 also as common multiples. But we should only consider the least once, so the LCM of \[{\text{3 and 10}}\] is 30.
We can also find LCM of two or more numbers by prime factorization method too.