Spoiler alert! Don't read on if you still want more time to find a quick solution to this type of puzzle. A 3-second method to show that these various numbers proposed are not perfect squares:
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All the numbers proposed so far have been divisible by 2 but not by 4. The unique prime factorization of any perfect square must consist of an even number of each prime factor. In this case we need only notice that there is 1 factor of 2, not an even number of 2s, therefore the number is not a perfect square.
How can we at a glance know that 60993193026 is divisible by 2 but not by 4?
2 divides x if and only if 2 divides the last digit of x: so if x ends with 0,2,4,6,8.
4 divides x if and only if 4 divides the number formed by the last 2 digits of x.
We see immediately that the long number above ends with 6 (divisible by 2) and its last 2 digits are 26: not divisibility by 4. Therefore the number is not a perfect square.
It is easy to prove that these above-mentioned divisibility rules are valid:
For 2: Write x = 10a + b, where b is the last digit of x, and a is the rest of the number. Since 2 divides 10, 2 divides 10a, and so 2 divides x if and only if x divides b.
For 4: Write x = 100a + b, where b is the number formed from the last two digits of x, and a is the rest of the number. Since 4 divides 100, 4 divides 100a, and so 4 divides x if and only if x divides b.
[Some steps skipped, but this gives the gist of the proof.]
Now, for your continued pleasure, is this a perfect square? 24 066 000 624 000
There are 2 quick ways to show that it isn't!
Enjoy!
Ken Caviness
Physics Department
Southern Adventist University
-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu on behalf of John Denker
Sent: Tue 15-Apr-08 2:47 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] perfect-square puzzler
Some people found yesterday's puzzler to be too easy.
I reckon what's easy for people on this list may not
be so easy for high-school students, but let's save
that discussion for another day.
In any case, here's another one: 60993193026.
Is that a perfect square, or not?
This number is shorter than the ones we played with yesterday.
-- Alice used her TI-84 and quickly came up with the right answer.
-- Bob used his TI-84 and quickly came up with the wrong answer,
with confidence, not realizing it was wrong.
-- Carol didn't use any kind of calculator at all, and came up
with the right answer, using even less time than the others did.
So ... how would you solve this? How confident are you in the
result? How would you explain/prove your method to someone who
wasn't familiar with it?
==========
I think puzzlers like this are valuable for several reasons:
*) For one thing, they lead students to re-evaluate what they know about
the number system. They were originally taught the number system by
rote, at an age when it would have been inappropriate to do otherwise.
Ditto for long multiplication. But at some point they need to re-learn
this stuff, paying attention to the principles involved. This includes
realizing that place-value in general and long multiplication in particular
are firmly rooted in the axioms of addition and multiplication, with a
little bit of modular arithmetic thrown in.
The famous puzzle
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MONEY
also advances this goal.
*) In September, there are plenty of students who need to be reminded
what FOIL means. (I realize that these mathematical puzzlers are more
suited to September than April. I assume folks on the list are able to
file them away and pull them out when needed.)
*) Another thing is that yesterday's puzzler rewards a certain amount of
backwards thinking. The underlying equation is y=x^2, and the question
tends to focus attention on y, but a systematic explanation benefits from
attention to the x values. Backwards thinking is a valuable skill. Some
students pick it up on their own, but others need to be taught. It is
over-rewarded by certain types of multiple-guess tests, but it can be
nicely applied in other situations, which is the point to be emphasized
here.