Find the smallest possible integer value of x such that √540x is a whole number.

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# Question

# Answer

This question is similar to the previous questions you posted but phrased in a different way. “Find the smallest possible integer value of x such that √540x is a whole number” is similar to “Find the smallest possible integer value of x such that 540x is a perfect square”. **Either way, we need to find a value x so that when it is multiplied by 540, it can be perfectly squared.**

540 = 2 x 270

= 2 x 2 x 135

= 2 x 2 x 3 x 45

= 2 x 2 x 3 x 3 x 15

= 2 x 2 x 3 x 3 x 3 x 5

We need one more 3 and one more 5 to make all the factors a perfect square, so…

3 x 5 = **15**

**Check:**

540 x 15 = 8,100

8,100 = 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5

= (2 x 3 x 3 x 5)^{2}

= 90^{2}

√8,100 = 90

Thanks I get it now!!!

Really appreciate your guidance!😄

Hello,

This problem is really easy, we need to prove that 540x is equal to something*something.

We use decomposition to prove that:

540x= 270*2*x

= 90*3*2*x

= 30*3*3*2*x

= 15*2*3*3*2*x

According to the multiplication rules we can write:

540x= 2*2*3*3*15*x

So:

√540x=2*3*√15x

So if we want √540x to be a whole number with x as smallest as possible, there is only x=15 that matches.

thanks 🙂

Hope it will help understand how to proceed

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