Question

LCM of two numbers x and y is 720 and the LCM of numbers 12x and 5y is also 720. The number y is:
A. 180
B. 144
C. 120
D. 90

Answer 1

Hint: We have been given two unknown numbers ‘x’ and ‘y’ along with their LCM as 720 which is also the LCM for ‘12x’ and ‘5y’. This will tell us that 12 is a factor to y and 5 to x. Thus, their LCM will be a factor to the LCM of and x and y. Then we will divide 720 by the LCM of 12 and 5 and that will give us another factor. The then obtained factor can be considered as a factor to both x and y. Thus, this will give us all the factors of y. By multiplying them, we will get the required value of y.

Complete step-by-step answer:
Now, we have been given two unknown numbers x and y. We also have been given that their LCM is 720. Also, the LCM of 12x and 5y is 720.
Thus, the LCM of x and y and the LCM of 12x and 5y are the same.
Since, the LCM of x and y and the LCM of 12x and 5y is same, i.e. 720, the LCM of 12x and y and the LCM of x and 5y is also the same, i..e. 720 since 12x and 5y are the multiples of x and y.
Now, after multiplying x by 12, the LCM doesn’t change. This means that 12 is already a factor of y. This is explained as follows:
Now, let us assume that y doesn’t have 12 as a factor. We know that LCM stands for least common multiple. Thus, 720 should have both 12x and y as its factor. 720 is also the LCM of x and y. This means that if x is multiplied with any number which is not a factor of y, the LCM will be multiplied by the same number too. But when x is multiplied by 12, the LCM remains the same. Thus, our contradiction is wrong.
Therefore, we can say that 12 is a factor of y.
Now, after multiplying y by 5, the LCM doesn’t change. This means that 5 is already a factor of x (this can be explained in the same way as we explained how 12 is a factor of y).
Thus, one of the factors of y is 12 and that of x is 5.
Now, the factors of x and y will also be factors to the LCM of x and y (as x and y are multiple of their factors and their LCM is also their multiple, hence it will be a multiple of the factors of x and y).
Thus, 12 and 5 will be factors of the LCM of x and y, i.e. 720.
Now, 12 and 5 are co-prime numbers (co-prime numbers are the numbers which don’t have any common factor except for 1). Thus, their product will also be a factor to 720.
Dividing 720 by the product of 12 and 5 we get:
$\begin{align}
  & 720\div \left( 12\times 5 \right) \\
 & \Rightarrow 720\div 60 \\
 & \Rightarrow 12 \\
\end{align}$
Thus, the required remaining factor of the LCM is 12.
From the amount of information that has been mentioned to us in the question, we cannot say for sure if this remaining 12 is a factor of only x or only y or both x and y. So, we can assume that it is a factor to both of them as it is the factor of their LCM.
Thus, 12 is a factor to both x and y. We already found out earlier that y has 12 as one factor and x has 5.
Thus, the factors of y are 12 and 12 (we wrote them like this because both the 12’s here are distinct factors of y).
Thus, we can write y as:
$\begin{align}
  & y=12\times 12 \\
 & \Rightarrow y=144 \\
\end{align}$
Thus, the value of y is 144.

So, the correct answer is “Option B”.

Note: When we divided 720 with the product of 12 and 5, we did so because 12 and 5 are co-prime. If they weren’t co-prime, 720 would have been divided by their LCM. This is because the product of two non coprime numbers may have a factor repeated more times than it is repeated in the LCM. Thus, the product may or may not be a factor to it. Thus, we should always take the LCM of the known factors. In the case of co-prime numbers, their LCM is equal to their product only, so doing their product is fine.
For example: if we take the LCM as 80 and two non coprime factors as 5 and 10, then their product is $5\times 10=50$ which is not a factor of 80 and if we take two non co-prime factors as 4 and 10, then their product $4\times 10=40$ is a factor.