Square Numbers End in 0 1 4 5 6 9Date: 10/15/2002 at 13:27:31 From: Holly Gray Subject: Square numbers Why do all square numbers end in 0 1 4 5 6 9? Yours sincerely, Holly Date: 10/15/2002 at 14:13:31 From: Doctor Ian Subject: Re: Square numbers Hi Holly, Here's how I would think about it. Suppose I have a number like 2234, and I want to find the square. I can multiply the number by itself, 11 11 <-- carries 2234 * 2234 ------ 8936 6702 ... ----- ...6 Now, note that what ends up in the ones column depends _only_ on the digits in the ones column. None of the other digits can possibly affect it. So all of the following will have exactly the same thing in the ones column: 4 * 4 = 16 14 * 14 = ...6 214 * 214 = ...6 5214 * 5214 = ...6 and so on. So all we have to do is look at what happens when we square single digits: 0 * 0 = 0 ends in 0 1 * 1 = 1 1 2 * 2 = 4 4 3 * 3 = 9 9 4 * 4 = 16 6 5 * 5 = 25 5 6 * 6 = 36 6 7 * 7 = 49 9 8 * 8 = 64 4 9 * 9 = 81 1 So, unless we invent some new digits, there's no way to get a square to end in anything but 0, 1, 4, 5, 6, or 9. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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