Question

L.C.M of 3, 40 is 120. L.C.M of 5, 24 is 120. L.C.M of 15, 8 is 120. Find another two pairs such that their L.C.M is 120.

Answer 1

Hint:
We will first find all the factors of 120. Then we will list all the factors. Using these factors, we will find combinations of pairs of numbers whose L.C.M is 120. LCM or Least Common Multiple is used to find the smallest common multiple among two or more numbers.

Complete step by step solution:
We have to find two pairs of numbers whose L.C.M is 120. Let us first find all the factors of 120. We know that every number has 1 and the number itself as its factors. So, 1 and 120 are factors of 120.
Since 120 is an even number, 2 is also a factor.
Also, $120 = 2 \times 60$.
So, 60 is also a factor.
We know that 3 is a factor of 12. So, it is a factor of 120 as well.
Also, $120 = 3 \times 40$.
Thus, 40 is also a factor.
Next, 4 is a factor of 12. So, it is a factor of 120 as well.
Now, $120 = 4 \times 30$.
This gives us 30 as a factor of 120.
Since 120 has 0 as its unit’s digit, 120 has 5 and 10 as its factors.
$120 = 5 \times 24{\text{ and }}120 = 10 \times 12$.
So, 24 and 12 are also factors.
Now, 6 is a factor of 12. This makes 6 a factor of 120 as well.
Also, $120 = 6 \times 20$.
So, 20 is also a factor.
Since 2 and 4 are factors of 120, $2 \times 4 = 8$ is also a factor of 120.
Also, $120 = 8 \times 15$.
So, 15 is also a factor of 120.
Thus, the factors of 120 are $1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120$. Using these factors, we will find the list of all pairs of numbers whose L.C.M is 120. The pairs are as follows:
\[ \left( {1,120} \right){\text{ }},{\text{ }}\left( {2,120} \right){\text{ }},{\text{ }}\left( {3,120} \right){\text{ }},{\text{ }}\left( {4,120} \right){\text{ }},{\text{ }}\left( {5,120} \right){\text{ }},{\text{ }}\left( {6,120} \right){\text{ }},{\text{ }}\left( {8,120} \right){\text{ }},{\text{ }}\left( {10,120} \right){\text{ }},{\text{ }}\left( {12,120} \right){\text{ }},{\text{ }}\left( {15,120} \right),{\text{ }} \\
  \left( {20,120} \right),{\text{ }}\left( {24,120} \right),{\text{ }}\left( {30,120} \right){\text{ }},{\text{ }}\left( {40,120} \right){\text{ }},{\text{ }}\left( {60,{\text{ }}120} \right){\text{ }},{\text{ }}(3,40){\text{ }},{\text{ }}(5,24){\text{ }},{\text{ }}(6,40){\text{ }},{\text{ }}(8,15){\text{ }},{\text{ }}(8,30){\text{ }},{\text{ }}(8,60){\text{ }},{\text{ }} \\
  (10,24){\text{ }},{\text{ }}(15,24){\text{ }},{\text{ }}(15,40){\text{ }},{\text{ }}(20,24){\text{ }},{\text{ }}(24,30){\text{ }},{\text{ }}(24,40){\text{ }},{\text{ }}(24,60),{\text{ }}(30,40). \\ \]

Using the above list, we can take $\left( {8,30} \right)$ and $\left( {15,40} \right)$ as our required pairs.

Note:
LCM of any two numbers should be divisible by both the numbers. The above pairs have been found by combining two factors from the list of factors whose L.C.M is 120. We have used trial and error methods i.e., we have selected two numbers and checked if their L.C.M is 120. Larger numbers will have longer lists.