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"I noticed that all the perfect squares are either a multiple of 4, or they're one more than a multiple of 4. Is that a rule or something?" ~Wes in South Dakota
Wes, I don't know if it's a rule, exactly, but it's definitely a true statement. Every perfect square is either a multiple of four, or one more than a multiple of four. Would you like me to prove it?
Every integer is either even or odd. That means it's either a multiple of two, or it isn't. If it's a multiple of two, we can write it like this:
If it's NOT a multiple of two, than it must be one more than a multiple of two. That means we can write it like this:
2n + 1
Every integer can be expressed either as 2n or as 2n + 1.
Therefore, every perfect square can be expressed as either (2n)2 or (2n + 1)2.
Now, if you've studied a bit of Algebra, you hopefully know how to simplify both of those expressions:
(2n)2 = 22n2 = 4n2
(2n + 1)2 = (2n + 1)(2n + 1) = 2n(2n + 1) + 1(2n + 1) = 4n2 + 2n + 2n + 1 = 4n2 + 4n + 1 = 4(n2 + n) + 1
Notice the result! The first possibility gives us a multiple of four, and the second one gives us one more than a multiple of four! And since that covers every perfect square, we can conclude that all perfect squares are either a multiple of four or one more than a multiple of four!