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**JaneFairfax****Member**- Registered: 2007-02-23
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ganesh wrote:

Non primes are natural numbers that are not prime, also known as composite numbers.

No, not all natural numbers that are not prime are composite numbers!

A composite number is a number that is a product of two or more prime numbers. The number 1 is not a composite number. However, the number 1 is a natural number that is not prime!

So, do you want composite numbers (which will exclude 1), or do you want non-prime natural numbers (which will include 1)?

*Last edited by JaneFairfax (2007-10-31 05:34:07)*

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**ganesh****Moderator**- Registered: 2005-06-28
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Jane Fairfax, I wanted non primes only.

Sometimes they are also referred to as composite numbers. I was just trying to be clear in my question what non primes are. Okay, I shall make it non primes alone. Thanks for correcting me.

Sometimes, in my over enthusiasm I make mistakes. Thanks for correcting me, Jane Fairfax

Character is who you are when no one is looking.

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**JohnnyReinB****Member**- Registered: 2007-10-08
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well, since no one answered this question I'm looking at...

*"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted" *

Nisi Quam Primum, Nequequam

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**ganesh****Moderator**- Registered: 2005-06-28
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Johnny, great work! You got it right! **Congratulations!!!**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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Replies to #33, #37 and#38

mathsyperson is absolutely correct! All the answers you posted are bull's eye.

And you remember right. Long back, this question (#38) was posted by me.

**Well done, mathsy! Keep it up!**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#41. Create a 4X4 Magic Square using the numbers 1-16 without referring. If you can do it in less than five minutes, thats a very good performance.

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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$42. We all know the product of a number is maximum when the number is divided into equal parts.**. The question is, how many equal parts?**

*PS:- Time limit :- 15 daysAfter 15 days, if there are no replies or no CORRECT replies, I shall post the solution in EULER Avenue!*

Character is who you are when no one is looking.

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**Identity****Member**- Registered: 2007-04-18
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**ganesh****Moderator**- Registered: 2005-06-28
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Solution to #42.

Very good, Identity!!!!!

You are correct! You hit the bull's eye!

*I shall wait for some time, say 15 days for the proof now.*

Character is who you are when no one is looking.

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

*Last edited by TheDude (2007-11-09 03:16:14)*

Wrap it in bacon

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 13,576

#42,

TheDude,

Solution is right! Proof is perfectly acceptable, right!

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Surely in that proof, x needs to be restricted to the naturals? That complicates things a bit.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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mathsyperson,

Yes. x is restricted to Natural numbers.

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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Solution to #42.

Outstanding, TheDude! I wholeheartedly accept your proof.

The solution is,

into as many parts as when the parts are closest to e.

The proof, the solution are given supra.

Example:- Question :Divide 10 into equal parts such that the produce is the maximum.

Solution :

Two parts

10/2 = 5

Product 5x5=25.

Three parts

10/3=3.3333=10/3 approximately.

Product 10/3 x 10/3 x 10/3 = 1000/27. = 37 approximately

Four parts

10/4=2.5

Product = 10/4 x 10/4 x 10/4 x 10/4 = 10000/256 = 39.0625

Five parts

10/5=2

Product = 2x2x2x2x2 = 32.

Six parts

10/6 = 1.66666

Product = 10/6 x 10/6 x 10/6 x 10/6 x 10/6 x 10/6 = 1000000/46656 = 21.43 approximately.

The number of parts close to e, that is the natural logarithm base, of value 2.71828 roughly, are 2.5 and 3.33333 when divided into 4 equal parts and 3 equal parts respectively. Of the two of them, it is clear 2.5 is closer to 2.71828 than 3.3333.

**Therefore, the product is the greatest when the number 10 is divided into 4 equal parts. As given above, the product is 39.0625, visibly, which is the highest of all.**

The number e appears where we expect it least in Mathematics. This question and proof is intended to show the fact. To a novice or a layman, it would be immensely suprising to know that an irrational number would have anything to do with the seemingly simple question. As a matter of fact, the obvious reply to the question would have been 2 parts. Mathematics is a subject which requires a degree of certainity to a much greater level. I am pleased that the question has been solved in such a short duration. **Commendabl performance, TheDude and Identity!**

Character is who you are when no one is looking.

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**Identity****Member**- Registered: 2007-04-18
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Thanks, but it was just an curiosity I remembered from somewhere, before seeing TheDude's proof I wouldn't have a clue where to start!

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**ganesh****Moderator**- Registered: 2005-06-28
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#43. I have 1000 one cent coins. These have to be packed in ten plastic bags in such a way that I should be able to give away any number of cents from 1 to 1000 in bags alone. For example, after I have put the 1000 one cent coins in bags, I may be required to give away 1 cent, or 100 cents or 433 cents etc. The number may vary from 1 to 1000. I should not open the bags once the cents are put in the plastic bags and I have marked how many cents each bag contains. I should give away bags only. How would I distribute the 1000 on cent dimes in the ten plastic bags?

Character is who you are when no one is looking.

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**TheDude****Member**- Registered: 2007-10-23
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*Last edited by TheDude (2007-12-12 01:23:13)*

Wrap it in bacon

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**ganesh****Moderator**- Registered: 2005-06-28
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Solution to #43.**Excellent, TheDude!!! Keep it up!!!**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#44. A goat is tied to one corner of a square plot of side 12m by a rope 7m long. Find the area it can graze?

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#45. The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member ?

Character is who you are when no one is looking.

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**JohnnyReinB****Member**- Registered: 2007-10-08
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*"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted" *

Nisi Quam Primum, Nequequam

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**Identity****Member**- Registered: 2007-04-18
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**ganesh****Moderator**- Registered: 2005-06-28
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Solutions to #44 and #45:-

Excellent, JohnnyReinB and Identity!!!

Your answers are perfectly correct!!!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#46. A farmer built a fence around his 17 cows, in a square shaped region. He used 27 fence poles on each side of the square. How many poles did he need altogether?

Character is who you are when no one is looking.

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**JohnnyReinB****Member**- Registered: 2007-10-08
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*"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted" *

Nisi Quam Primum, Nequequam

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