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5 ways of factorizing (Posted on 2018-07-07) Difficulty: 3 of 5
What is the smallest number that can be shown as a difference between two squares in 5 different ways?

See The Solution Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Barring an oversight, the answer is 192

The number must have at least 10 factors to fit 5 instances of (a+b)(a-b).  Also, (a+b) and (a-b) will necessarily have the same parity as otherwise adding them leads to a contradiction.

Looking at increasingly larger numbers with at least 10 factors:

48 has 4 pairs of factors with same parity

80 has only 3 

96 and 160 have 4

192 has 5, (96,2),(48,4),(32,6),(24,8),(16,12).

Setting a=1/2 the sum and b=1/2 the difference of the factors gives solutions (49,47),(26,22),(19,13),(16,8),(14,2).


  Posted by xdog on 2018-07-07 13:10:32
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