Question

& Find the smallest square number that is divisible by each of the numbers 8,15,20,30
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Answer 1

Answer :

For finding the smallest square number that is divisible by each of the numbers 8,15,20,30 we need to find the LCM .

I find it see the attachment.

Here, prime factors 2, 3, and 5 do not have their respective pairs.

Therefore, 120 is not a perfect square.

Therefore, 120 should be multiplied by 2 × 3 × 5, i.e. 30, to obtain a perfect square.

Hence, the required small square number is 120 × 2 × 3 × 5 = 3600.

Answer 2

Question

Find the smallest square number that is divisible by each of the numbers 8, 15, 20 and 30.

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Answer

Step 1: Find the LCM of 8, 15, 20 and 30.

$\begin{array}{r | l} 2 & 8, 15,20,30\\ \cline{2-2} 2 & 4,15,10,15\\ \cline{2-2} 3 & 2,15,5,15 \\ \cline{2-2} 5 & 2,5,5,5\\ \cline{2-2} & 2, 1,1,1 \end{array}$

LCM of 8, 15, 20 and 30 = 2×2×2×3×5 = 120

Step 2: Pair the prime numbers.

$(2\times 2) \times 2 \times 3 \times 5$

Step 3: Multiply 30 (2×3×5) to 120 so these numbers can have pairs and be a perfect square.

120×30 = 3600

Step 4: Prime factorize 3600 for verification if it is a perfect square.

$\begin{array}{r | l} 2 & 3600 \\ \cline{2-2} 2 & 1800 \\ \cline{2-2} 2 & 900 \\ \cline{2-2} 2 &450 \\ \cline{2-2} 3 & 225\\ \cline{2-2} 3 & 75 \\ \cline{2-2} 5 &25\\ \cline{2-2}&5\\ \end{array}$

Step 5: Pair the prime numbers.

$(2\times 2)\times (2 \times 2) \times (3\times 3)\times (5\times 5)$

Step 6: Take one number from each pair and multiply them to obtain the square root.

$\sqrt{3600} = 2\times 2\times 3\times 5$

$\sqrt{3600} = 60$

∴The least number that is divisible by 8, 15, 20, 30 is 3600.

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Additional Information

What is a square number?

When we multiply any number two times to itself the product is known as the square number.

What is square root?

When we find the two numbers that were multiplied together to obtain the product is known as the square root.

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