Question

The least number which is a perfect square and is divisible by each of 16, 20 and 24 is:
A.240
B.1600
C.2400
D.3600

Answer 1

Hint: Here we will first find the LCM of these three numbers. Then we will multiply the number that are not in pair in the factorization of the LCM because for a number to be a perfect square each factor should be in pair. We will simplify it further to get the required answer.

Complete step-by-step answer:
We will first find the LCM of these three numbers using the method of prime factorization.
Now, we will first express each number in terms of the product of its prime factors.
To write the prime factors, we should always start with the smallest prime number i.e. 2 and check divisibility. If the number is divisible by the prime number, then we can write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by that prime number. Then we take the quotient and we will repeat the same process. This process is repeated until we get the quotient as 1.
Now, we will find the factors of the first number 84.
We know that 16 is an even number, so it can be written as $2\times 8$ i.e.
$16=2\times 8$
Now, we will further break the number 8 into its factors.
$\Rightarrow 16=2\times 2\times 2\times 2$
Similarly, we will find the factors of the second number 20.
We know that 20 is an even number, so it can be written as $2\times 10$ i.e.
$20=2\times 10$
Now, we will further break the number 10 into its factors.
$\Rightarrow 20=2\times 2\times 5$
Again we will find the factors of the third number 24.
We know that 24 is an even number, so it can be written as $2\times 12$ i.e.
$24=2\times 12$
Now, we will further break the number 12 into its factors as we can see that it is a multiple of 2 and 3.
Therefore, we can write it as
$\Rightarrow 24=2\times 2\times 2\times 3$
Now, to find the LCM, we need to multiply all the prime factors that has occurred the maximum number of time in either of the numbers. So, LCM of 16, 20 and 24 is a product of four 2, one 3 and one 5. As we can see that 2 have occurred four times in 24, 3 is occurring a maximum number of 1 time and 5 are occurring maximum number of one time.
Therefore, $LCM\left(16,20,24\right)=2\times 2\times 2\times 2\times 3\times 5$
On multiplying the number, we get
$\Rightarrow LCM\left(16,20,24\right)=240$
We know that for a number to be a perfect square each factor should be in pair of two or in even numbers we can see that 3 and 5 are not in pair of two. So to get the number, we will multiply 3 and 5 to the LCM obtained.
Now we will find the least number that is perfect square and it is divisible by all these given numbers.
The required least number $=2\times 2\times 2\times 2\times 3\times 3\times 5\times 5$
On multiplying the numbers, we get
$\Rightarrow$ The required least number $=3600$
Hence, the correct option is option D.

Note: Sometimes we get confused between factors of a number and multiples of a number. A factor is defined as a number which divides the given number completely but the multiple is defined as a number that is completely divided by the given number. Let’s take the number 10 as an example. We know that the factors of number 10 are 1, 2, 5 and 10 but the multiples of number 10 are 10, 20, 30, 40, and so on.