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#101 2007-10-31 04:23:47

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: *** Problems

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! shame

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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#102 2007-10-31 05:09:13

ganesh
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Registered: 2005-06-28
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Re: *** Problems

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 smile


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#103 2007-11-03 22:09:26

JohnnyReinB
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Registered: 2007-10-08
Posts: 453

Re: *** Problems

well, since no one answered this question I'm looking at...


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#104 2007-11-04 00:57:11

ganesh
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Re: *** Problems

Johnny, great work! You got it right!
Congratulations!!!


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#105 2007-11-06 03:07:00

ganesh
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Posts: 14,421

Re: *** Problems

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!


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#106 2007-11-06 03:13:17

ganesh
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Re: *** Problems

#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.


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#107 2007-11-08 19:10:09

ganesh
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Re: *** Problems

$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 days
After 15 days, if there are no replies or no CORRECT replies, I shall post the solution in EULER Avenue!


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#108 2007-11-09 00:53:08

Identity
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Registered: 2007-04-18
Posts: 934

Re: *** Problems

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#109 2007-11-09 01:57:56

ganesh
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Re: *** Problems

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.


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#110 2007-11-09 03:15:42

TheDude
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Registered: 2007-10-23
Posts: 361

Re: *** Problems

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


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#111 2007-11-09 03:30:17

ganesh
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Registered: 2005-06-28
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Re: *** Problems

#42,
TheDude,
Solution is right! Proof is perfectly acceptable, right!


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#112 2007-11-09 05:46:09

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: *** Problems

Surely in that proof, x needs to be restricted to the naturals? That complicates things a bit.


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#113 2007-11-09 20:21:24

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

mathsyperson,
Yes. x is restricted to Natural numbers.


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#114 2007-11-09 20:42:11

ganesh
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Registered: 2005-06-28
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Re: *** Problems

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!


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#115 2007-11-09 21:16:49

Identity
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Registered: 2007-04-18
Posts: 934

Re: *** Problems

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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#116 2007-12-11 21:49:35

ganesh
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Posts: 14,421

Re: *** Problems

#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?


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#117 2007-12-12 01:20:19

TheDude
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Registered: 2007-10-23
Posts: 361

Re: *** Problems

Last edited by TheDude (2007-12-12 01:23:13)


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#118 2007-12-12 02:03:13

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

Solution to #43.
Excellent, TheDude!!! Keep it up!!!


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#119 2007-12-12 20:21:26

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

#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?


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#120 2007-12-12 20:23:23

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

#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 ?


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#121 2007-12-14 16:24:52

JohnnyReinB
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Registered: 2007-10-08
Posts: 453

Re: *** Problems


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

Nisi Quam Primum, Nequequam

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#122 2007-12-14 16:55:57

Identity
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Registered: 2007-04-18
Posts: 934

Re: *** Problems

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#123 2007-12-15 17:09:53

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

Solutions to #44 and #45:-
Excellent, JohnnyReinB and Identity!!!
Your answers are perfectly correct!!!


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#124 2007-12-15 17:13:05

ganesh
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Registered: 2005-06-28
Posts: 14,421

Re: *** Problems

#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?


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#125 2007-12-15 22:36:12

JohnnyReinB
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Registered: 2007-10-08
Posts: 453

Re: *** Problems


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

Nisi Quam Primum, Nequequam

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