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Make it solvable (Posted on 2005-05-05) Difficulty: 4 of 5
a, b, and x are positive integers such that

sqrt(a) + sqrt(b) = sqrt(x)

How many possible values of x less than or equal to 1000 are there?

See The Solution Submitted by Jer    
Rating: 3.6667 (3 votes)

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Hints/Tips First thoughts... ...(spoiler?) | Comment 1 of 16

It seems to me that there are a rather large number of valid solutions.  a, b, and x must all be squares, and x can be no larger than 1000, so the sqrt's can add up to no more than 31 (31^2 = 961, 32^2 = 1024).

<pre>
sqrt        ^2
   1           1
   2           4
   3           9
   4          16
   5          25
   6          36
   7          49
.
.
.
  30         900
  31         961
</pre>

Now we just need to find all the values from that list where the numbers from the first column add up to no more than 31.  So a=1, b=1, x=4 (1+1=2) through a=225, b=256, x=961 (15+16=31).  I'm assuming that we're not looking for reversals of a and b, so if we already have sqrt(1)+sqrt(900)=sqrt(961) we don't later include sqrt(900)+sqrt(1)+sqrt(961).

If that is the case, then I count 345 unique solutions.


  Posted by Erik O. on 2005-05-05 18:51:48
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