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If x is a positive integer less than 100, how many values of x make sqrt(1 + 2 + 3 + 4 + x) an integer?

 Jun 8, 2020
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Think of all the perfect squares under 100.

 

There's 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

 

Now, we know that \(\sqrt{10 + x} = \text{Some integer}\), right?


Because x has to be positive, that rules out 1, 2, 4, and 9 to be the square.

 

It could be 16, 25, 36, 49, 64, 81, and 100.


That is 7 different numbers, which means 7 different values of x!

 Jun 8, 2020

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