Question

find the smallest square number that is divisible by 8,12,15 and 20

Answer 1

First let's find the LCM of 8,12,15,and 20.

8=2×2×2

12=2×2×3

15=3×5

20=2×2×5

So the LCM is 2×2×2×3×5=120

Here is 2,3,5, are not in pairs, we need to in pairs for smallest square number.

120×2×3×5=120×30=3600.

Hence the number is 2×2×2×2×3×3×5×5=3600.

3600 is the right answer.

Answer 2

Given: Four numbers 8, 12, 15 and 20

To find: The smallest square number that is divisible by given numbers

Solution: To find the required number, first we need to find the smallest number which is exactly divisible by the given numbers i.e. LCM.

Using prime factorization method:

8 = 2 × 2 × 2

12 = 2 × 2 × 3

15 = 3 × 5

20 = 2 × 2 × 5

LCM is the product of maximum frequencies of all the factors of given numbers.

LCM = 2 × 2 × 2 × 3 × 5 = 120

Now we need to find a multiple of 120 which is a perfect square.

We know that 120 × 30 = 3600 (which is a perfect square)

Also, we can observe that prime factorization of 120 lack 2, 3 and 5 to make pairs. If we multiply it by 2 × 3 × 5 i.e., 30, it will become a perfect square.

Hence, the smallest square number that is divisible by 8, 12, 15 and 20 is 3600.