Perfect Squares: n+125 and n+201Date: 01/21/2002 at 06:43:52 From: Lisa Subject: Perfect squares Find the smallest positive integer n so that n+125 and n+201 are both perfect squares. Date: 01/21/2002 at 11:12:18 From: Doctor Paul Subject: Re: Perfect squares The answer is 199. Here's how to do it: Can you make a list of perfect squares? 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 What values of n make n+125 a perfect square? To solve the above question, you need to subtract 125 from every number on the list of perfect squares that is greater than 125. This gives: 19 44 71 100 131 164 199 236 275 316 359 404 451 500 Now, what values of n make n+201 a perfect square? To solve the above question, you need to subtract 201 from every number on the list of perfect squares that is greater than 201. 24 55 88 123 160 199 240 283 328 375 424 To complete the problem, we're looking for the smallest number on both lists - i.e., the smallest number that makes both n+125 and n+201 a perfect square. Clearly, the answer is 199 since 199+125 = 324 = 18^2 and 199+201 = 400 = 20^2. I wanted to know what the next number was that makes n+125 and n+199 a perfect square so I wrote a little program and asked my computer to find it. I gave up at ten million... I hope this helps. Please write back if you'd like to talk about this more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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