 # Question

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

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
= 902

√8,100 = 90

Thanks I get it now!!!

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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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