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Kate kate_nepveu
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A quick puzzle

To make up for being cranky, a quick puzzle for you all from Car Talk

A student is asked if the following number is a perfect square: 334,912,740,121,562. Crusty knows the answer immediately. What does Crusty know?

I think it would be tedious to spoiler-protect comments, so if you want to respond without seeing other comments, go straight to the reply-only link.




The last digit of the square of a natural number is determined entirely by its own last digit, and none of the squares of single-digit numbers ends in 2.

Yup.

I suspect this is a much easier puzzle in text than out loud. When listening I zoned out somewhere in the millions.

(edited for spelling)

Edited at 2008-03-18 01:53 pm (UTC)

No perfect square ends in 2. All perfect squares have, as their ones digit, the square of the ones digit of the number they're squares of, and nothing between 0 and 9 ends in 2 when squared. So there's no number that when squared ends in a 2.

Crusty knows that the student is nuts. Who would care if a 15 digit number is a perfect square?

oh, you mean, why isn't it a square? well, others have gotten to it first...

Hey, it was a school assignment . . .

Then he shouldn't have been getting help from a clown now SHOULD he? (or she)

Crusty's answer is: if your house really has that many square feet in it, then six inches one way or the other don't matter.

Since if Google conversions haven't failed me, that's all the arable land in the world, pretty much, yeah!

Squares can't end in 2. Only the right most digit affect the last digit of it's square. 1->1, 2->4, 3->9, 4->(1)6, 5->(2)5, 6->(3)6, 7->(4)9, 8->(6)4, 9->(8)1,and 0->0.
The only possibilities are 0 1 4 5 6 9. No 2 3 7 or 8

Number theory is fun.

Perfect squares must end in 1, 4, 9, 6, or 5. 2 is not on that list.

Crusty knows that no number squared ends in 2.

What do I win?

The satisfaction of being right.