A Perfect Square

Mar 08, 2008

This is from the wild, wacky, wonderful world of mathematics and it was sent in by William Dubuvitz. Here it is:

One of the mechanics in the garage has a son in high school who is a very bright student and very good in mathematics and computer programming. He stopped by the garage one day after school and his father asked what he was doing in school and his son told him about his latest assignment.

He is supposed to write a computer program to handle very large numbers that could not be handled on a typical hand-held calculator. The teacher told the students to use that program to determine if a certain very large number is a perfect square. (What is a perfect square? A perfect square is a whole number or an integer that is arrived at by squaring another whole number. For example, 900 is a perfect square of 30; 196 is a perfect square of 14. 625 is the perfect square of 25. So there are no fractions, no decimals, no nothing. Just whole numbers allowed.)

Each student is assigned a particular number. This kid's number is 334,912,740,121,562. And the teacher wants to know if this is a perfect square.

His father says, "That's a big number!"

And then out of the inky shadows, who appears but Crusty! And he says, "Oh, your teacher gave you an easy number."

"She did?" said the kid.

"Oh yeah. I can give you the answer right now."

The question is, what did Crusty know?


RAY: Well, our number ends with the number two. And there are no perfect squares that end in or have a units digit of two. It's impossible.

TOM: Impossible indeed.

RAY: Why? Because the units digit of any perfect square is determined by the units digit of its square root. Take the number one, for example, square it and you get one. Take the number two and square it, it comes out four. The number three comes out nine. But there is no number that you can square and get a two at the end. That's what Crusty knew. Do we have a winner?

TOM: The winner this week is Diana Carzoli from West Des Moines, Iowa, and for having her answer selected at random from among the pile of correct answers that we got, Diana is going to get a 26-dollar gift certificate to the Shameless Commerce Division at cartalk.com with which she can get one of our CD collections like, The Best and Second Best of Car Talk.

RAY: Wow.

TOM: Congratulations to you, Diana baby!

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