Question

Find the smallest number by which 3024 must be divided so that the quotient is a perfect
square. Also find the square root of the perfect square so obtained.

Answer 1

Step-by-step explanation:

What is the smallest number by which 3,920 must be divided to make it a perfect square?

To solve this, you would need to start by finding the prime factorization of 3,920, which is 24∗72∗5 . You could expand 24 to (22)2 , which makes your other equation (22)2∗72∗5=3920 . You can simplify 22 to 4, which makes your equation 42∗72∗5=3920 . Here’s a short proof for a theorem we’re going to need to answer this question. We need to prove that z is an integer for all positive integers x and y, given sqrt(x2∗y2)=z . First, we can use the square root identity that states that sqrt(x∗y)=sqrt(x)∗sqrt(y) . Using that identity in our other equation gives us sqrt(x2)∗sqrt(y2)=z . Because sqrt(x2)=x , we can change our other equation to x+y=z . Because we were given that x and y are positive integers, the sum of two positive integers is a positive integer. Because z is the sum of x and y, two positive integers, z must be a positive integer. We have just proven that sqrt(x2∗y2)=z . Squaring both sides gives us x2∗y2=z2 . Because z is a positive integer, z2 must be a perfect square. In our other equation, we can use this theorem to show that 42∗72 is a perfect square. Therefore, x2∗5=3920 , where x2=42∗72 . Solving for x gives you x=28. You can substitute 28 in for x and make your other equation 282∗5=3920 . You can divide both sides by 5 to get 282=3920/5 . There! We have shown that 3920 divided by 5 is a perfect square. Therefore, 5 is the smallest number by which 3920 must be divided to make it a perfect square.

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Answer 2

Step-by-step explanation:

Step 1 ) first take the LCM

Step 2 ) make three pairs of each numbers

step 3) 2 and 7 will left multiple them

step 4 ) it will become 14

The answer is 14