Find the number!

mathaholic · 2012-12-06 21:35:26

Find the number with the clues given. You may need to use some math on the way. I start with a problem, and then when people answers, then I make another problem.
Good luck!
I.
My number is:
1) A prime number.
2) Half of ten.
What number is it?

anonimnystefy · 2012-12-06 21:46:28

Hi

mathaholic · 2012-12-06 23:58:40

Wow, anonimnystefy!
:):):):)

noelevans · 2012-12-07 12:36:41

How about the largest number less than 100 that can only be written one way as a difference of
squares?

bobbym · 2012-12-07 13:55:41

Hi;

noelevans · 2012-12-07 14:31:54

Yep! Privy to a bit of number theory, eh? Would make Fermat smile in his grave.

How about a base (other than two) for which one can write any positive number in terms of using no coefficients greater than one?

phrontister · 2012-12-07 14:57:43

Hi noelevans,

mathaholic · 2012-12-07 15:30:30

Whoops, bobbym! 97 is a prime number, but is 10/2=97?

noelevans · 2012-12-07 15:33:34

Hi phrontister!

That's using your noodle and your computer! Turns out to be a great combination for lots of problem
solving. I bet Gauss, Newton, Euler, etc. would have given a lot to have the computational power
that we have today. They might not have left us any math to discover!

One of the characteristics of primes larger than two is that there is only one way to write them as
a difference of squares. Composites always have at least two ways to do it. Each factorization of
a number greater than two corresponds to a difference of squares factorization. For example
15=1x15=8^2-7^2 and 15=3x5=4^2-1^2. 21=1x21=11^2-10^2 and 21=3x7=5^2-2^2.
So a number like 27 which has 2 factorizations has 2 ways to write it as a difference of squares.

Primes and perfect squares are neat, multifaceted numbers.

Have a blessed evening!

Edit: Oops! Limit the above discussion to odd non-square composite numbers. Even numbers and
odd perfect squares may have fewer difference of square factorizations, especially if we do
not allow zero as one of the numbers. Example 4=4x1=(5/2 + 3/2)(5/2 - 3/2) but these are
not integral. Also 4=2x2=(2+0)(2-0) won't do if we disallow zero.

For 81=9x9 we have no difference of square factorization if we disallow zero.
But 81=1x81=(41+40)(41-40) and 81=3x27=(15+12)(15-12).

for 9=1x9=(5+4)(5-4) but 9=3x3=(3+0)(3-0) won't work if zero is disallowed.
So the square of a prime has only one difference of squares factorization IF we
disallow zero.

Last edited by noelevans (2012-12-07 15:59:47)

bobbym · 2012-12-07 16:03:34

Hi;

How about a base (other than two) for which one can write any positive number in terms of using no coefficients greater than one?

There is the Zeckendorf numbers which use Fibonacci numbers as the base. The coefficients do not have to be other than 1 and 0. For instance:

Hi julianthemath;

Whoops, bobbym! 97 is a prime number, but is 10/2=97?

Your problem's answer is 5 , noelevans' problem has the answer of 97.

noelevans · 2012-12-07 17:30:57

Zeckendorf representations --- VERY INTERESTING! Thanks for pointing them out.

Actually I was just thinking of a base as a single integer >1. I've never seen a SET being a base.
Wiki points out that this property is characteristic of the Fibonacci numbers so no other set of
numbers has this property ascribed to the Fibonacci numbers.

My question is a bit tricky. Any positive integral base greater than 1 could satisfy the conditions I
mentioned. The coefficients could be negative integers (introduced by a many named Colson in
- --
1729) for example 481 = 1x1000 - 5x100 - 1x10 -9x1 = 1519 where the bars above the digits
mean subtract instead of adding. So 481 is written without any coefficients >1.

But if we require that the absolute value of the integral coefficients not be greater than 1 then
base 2 and one other base will satisfy this.

I had this message written earlier and I previewed it but forgot to submit it, so I lost it.

Live and learn!

phrontister · 2012-12-07 17:31:39

Hi noelevans,

Edit: Oops!

Glad I spotted that you'd spotted that. I was close to putting pen to paper about my failure in getting the factorization & difference of squares feature to work with 22 and 72, and was looking at testing some other evens and then more odds.

mathaholic · 2012-12-07 19:46:49

Got it, bobbym!

mathaholic · 2012-12-07 19:54:54

II.
My number is:
1) A 3-digit number.
2) A square number.
3) A composite number.
4) Divisible by 11.
What is it?

bobbym · 2012-12-07 21:19:54

Hi;

phrontister · 2012-12-07 22:17:16

Hi julianthemath,

mathaholic · 2012-12-07 22:47:47

Okay!
Two official answers are:
121
484
Bobbym, Phrontister, you are correct!

bobbym wrote:

484

phrontister wrote:

121

:):):):):):):):):):):):):):):):):):)

Mathegocart · 2012-12-08 04:58:59

121 smilies,right.....LOL

bobbym · 2012-12-08 10:37:07

Hi;

Bobbym, Phrontister, you are correct!

I like phrontister's answer better. Where that other gets his answer from I can only speculate.

phrontister · 2012-12-08 11:08:10

bobbym wrote:

Where that other gets his answer from I can only speculate.

9 darts into a dartboard?

bobbym · 2012-12-08 11:09:21

Maybe, but he probably would miss the board!

mathaholic · 2012-12-08 11:12:39

III.
My number is:
1) A 5-digit number.
2) Is ranged from 100^2-109^2.
3) Has a "25" at the end.
What is it?
Tell me the square number and the square root.

phrontister · 2012-12-08 11:12:58

I meant thump them in with a hammer...not throw them.

phrontister · 2012-12-08 11:14:46

Hi julianthemath,

Last edited by phrontister (2012-12-08 11:17:09)

bobbym · 2012-12-08 11:47:38

Hi;

I meant thump them in with a hammer...not throw them.

That would be more accurate.

Math Is Fun Forum

#1 2012-12-06 21:35:26

Find the number!

#2 2012-12-06 21:46:28

Re: Find the number!

#3 2012-12-06 23:58:40

Re: Find the number!

#4 2012-12-07 12:36:41

Re: Find the number!

#5 2012-12-07 13:55:41

Re: Find the number!

#6 2012-12-07 14:31:54

Re: Find the number!

#7 2012-12-07 14:57:43

Re: Find the number!

#8 2012-12-07 15:30:30

Re: Find the number!

#9 2012-12-07 15:33:34

Re: Find the number!

#10 2012-12-07 16:03:34

Re: Find the number!

#11 2012-12-07 17:30:57

Re: Find the number!

#12 2012-12-07 17:31:39

Re: Find the number!

#13 2012-12-07 19:46:49

Re: Find the number!

#14 2012-12-07 19:54:54

Re: Find the number!

#15 2012-12-07 21:19:54

Re: Find the number!

#16 2012-12-07 22:17:16

Re: Find the number!

#17 2012-12-07 22:47:47

Re: Find the number!

#18 2012-12-08 04:58:59

Re: Find the number!

#19 2012-12-08 10:37:07

Re: Find the number!

#20 2012-12-08 11:08:10

Re: Find the number!

#21 2012-12-08 11:09:21

Re: Find the number!

#22 2012-12-08 11:12:39

Re: Find the number!

#23 2012-12-08 11:12:58

Re: Find the number!

#24 2012-12-08 11:14:46

Re: Find the number!

#25 2012-12-08 11:47:38

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