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# What's the smallest value of the natural number x(x ≠ 0), so that 280x is a perfect square?

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- rotchmLv 72 years agoBest Answer
280 = 2^3 * 5 * 7. You want that

2^3 * 5 * 7* x = n²

For this, all powers on the LHS bust be multiples of 2 [why?].

So you need an extra 2 to make 2^4, an extra 5 to make 5² and an extra 7 to make 7² .

Recap: An extra 2 & 5 & 7.

So, whats the answer for x?

Done!

- DixonLv 72 years ago
You need to make it so that 280x has paired prime factors, then it will be a perfect square.

280 = (2 x 2) x 2 x 5 x 7 which has one pair of twos but everything else is unpaired.

So if you put in another 2, 5 and 7 it will become square.

Thus x = 70

Could you check the comment I made on rotchm's answer? And thanks for helping.

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Yes, 280*280 is a perfect square. But as you notice, its not the *smallest* as requested in the question.