What is the least number which is exactly divisible by 8 9 12 15 and 18 and also a perfect square?

Answer

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Hint: We will first find the LCM of the given numbers, 15,18 and 25. Then we will check whether the number is a perfect square or not. If the resultant number is not a perfect square, then take a multiple of it, which is a perfect square.

Complete step-by-step answer:

Here, we start the question by finding the LCM of 15,18 and 25.LCM of 15,18 and 25 will give us the smallest common multiple of these numbers. Hence, that number will be divisible by all the given numbers.Let us first write the factors of each given number.The factors of 15 are $3 \times 5$The factors of 18 are $2 \times 3 \times 3$Similarly, the factors of 25 are $5 \times 5$.Next, write the LCM of the three numbers using their factors.LCM is calculated by multiplying each factor the maximum number of times it comes in each given number.That is, LCM of 15,18 and 25 is$2 \times 3 \times 3 \times 5 \times 5$This can also be written as, $2 \times {3^2} \times {5^2}$The above number will not make a perfect square as 2 is not multiplied with itself.But, we want to find the number which is a perfect square.Hence, multiply the number by 2.Then, we will get,${2^2} \times {3^2} \times {5^2} = 900$,which is a perfect squareThus, 900 is divisible by 15,18 and 25 and is perfect square.Hence, option D is the correct answer.

Note:- For finding the LCM, we can also use the common division method.

Hence, the LCM of 15,18 and 25 is $2 \times {3^2} \times {5^2}$Also, a perfect square is those numbers which are the product of the same integers.

Solution:

We will be using the concept of LCM(Least Common Multiple) to solve this.

To determine the greatest 3-digit number exactly divisible by 8, 10, and 12, we need to find the LCM of the given numbers.

Let's find the LCM of 8, 10, and 12 as shown below.

LCM of 8, 10, and 12 will be equal to 2 × 2 × 2 × 3 × 5 = 120

As we know that the greatest 3-digit number is 999, we will now divide 999 by the obtained LCM that is 120.

Since the remainder is not 0, hence we subtract 39 from 999.

999 - 39 = 960 which is a multiple of 120 and is exactly divisible.

Now,

120 × 8 = 960 and 120 × 9 = 1080.

We see that 120 × 9 = 1080 is a 4-digit number but, we need the greatest 3-digit multiple.

Therefore, the greatest 3-digit multiple of 120 is 120 × 8 = 960 i.e, 960.

Hence, the greatest 3-digit number exactly divisible by 8, 10, and 12 is 960.

You can also use the LCM Calculator to solve this.

NCERT Solutions for Class 6 Maths Chapter 3 Exercise 3.7 Question 5

Summary:

The greatest 3-digit number exactly divisible by 8, 10, and 12 is 960.

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