Problems and Solutions

Jai Ganesh · 2005-12-14 17:26:26

Problem # k + 63

Show that any prime number other than 2 can be expressed as the difference of two squares, where each square is an integer squared.

John E. Franklin · 2005-12-15 16:45:59

Last edited by John E. Franklin (2005-12-15 16:54:15)

Ricky · 2005-12-15 17:11:58

krassi_holmz · 2005-12-16 20:58:20

I think Ricky is right. Here's a direct proof:
Let
u^2-v^2 = p
We'll find u and v by p:
u^2-v^2 = p
=> (u+v)(u-v) = p, but p is prime, so his divisors are only 1 and p, so:
u+v=p && u-v=1
=> u=v+1; 2v+1=p; v= (p-1)/2, but p is odd, so v is integer.
So:
v= (p-1)/2
u=1 + ((p-1)/2)= (p+1)/2.

Last edited by krassi_holmz (2005-12-16 21:00:23)

Jai Ganesh · 2005-12-18 16:47:04

Correct, John, Ricky and krassi_holmz!

Problem # k + 64

For what value of 'n' will the remainder of 351^n and 352^n be the same when divided by 7?

krassi_holmz · 2005-12-18 18:04:40

Last edited by krassi_holmz (2005-12-18 18:07:55)

Jai Ganesh · 2005-12-18 22:58:39

Correct! Well done, krassi_holmz

Problem # k + 65

The difference between a number and two-thirds of its value is the sum of the digits of the original number. How many digits can this number have?

mathsyperson · 2005-12-19 00:13:34

Last edited by mathsyperson (2005-12-19 00:13:44)

Ricky · 2005-12-19 05:30:29

Something I just noticed about k+63. The same can be said for all odd numbers, no?

mathsyperson · 2005-12-19 07:27:00

True.
The difference of 2 squares is (n+1)² - n² = n² + 2n + 1 - n² = 2n+1.
Odd numbers are written as 2n+1 by definition, so that proves it.

Jai Ganesh · 2005-12-19 16:21:47

Well done, Mathsyperson!

Problem # k + 66

If 5^m = 35^6 = 7^n, what is the value of (mn)/(m+n)?

Jai Ganesh · 2005-12-20 16:08:41

Problem # k + 67

The Arithmetic Mean of two numbers is 25 and their Geometric Mean is 20. What is their Harmonic Mean?

Jai Ganesh · 2005-12-21 16:16:37

Problem # k + 68

A sports field consists of a rectangular region with semi-circular regions projecting at the two opposite sides. If the perimeter of the filed is 1,000 meters, find the area of the largest possible field.

Jai Ganesh · 2005-12-21 17:59:25

Problem # k + 69

What is the least value of n such that n!>10^n?

mathsyperson · 2005-12-22 02:20:27

Last edited by mathsyperson (2005-12-22 02:20:37)

John E. Franklin · 2005-12-22 14:16:24

ganesh, do you have a list of the problems
that are unsolved? That way I can go back
easier and solve some old ones.

Jai Ganesh · 2005-12-22 16:41:32

Well done, mathsyperson!
All of your solutions are correct.
The problem on AM, GM and HM was meant to be solved knowing AM*HM=GM²

Solution to Problem # k + 66

Problem : If 5^m = 35^6 = 7^n, what is the value of (mn)/(m+n)?

Solution : Let 5^m=(5*7)^6=7^n=k
5^m=k, therefore, 5 = k^(1/m)
7^n=k, therefore, 7 = k^(1/n)
35^6=k, therefore, 35=k^(1/6)
5 x 7 = 35,
therefore, k^(1/m)*k^(1/n)=k^(1/6)
k^[(m+n)/mn]= k^(1/6)
Therefore, (m+n)/mn = 1/6
(Bases are equal, therefore exponents are equal).
Hence, mn/(m+n)=6.

To John E. Franklin:- I shall soon make a list of unsolved problems.

Problem # k + 70

Give the first list of fifteen consecutive non-prime numbers.

Problem # k + 71

Prove that the only set of triplet primes is (3,5,7).
Triplet primes are three consecutive odd numbers which are all prime.
As a rule, (2,3,5) is not included.

krassi_holmz · 2005-12-23 06:15:38

Last edited by krassi_holmz (2005-12-23 06:35:26)

krassi_holmz · 2005-12-23 06:23:51

Here's plot of the function Prime[i+1]-Prime[i] up to 1000.
Actually, it's Prime[Floor[i+1]]-Prime[Floor[i]]

krassi_holmz · 2005-12-23 06:29:40

Last edited by krassi_holmz (2005-12-23 06:37:12)

Ricky · 2005-12-23 06:46:38

I worked up my own proof, but krassi beat me to it. I figure I'd post it anyways.

Last edited by Ricky (2005-12-23 07:48:33)

krassi_holmz · 2005-12-23 09:10:59

Well done, Ricky!
Your proof is very good.
And I haven't beated you because you was able to give another good proof.

Jai Ganesh · 2005-12-23 16:08:54

Well done, krassi_holmz and Ricky!
The solutions to both the problems are correct.
The two proofs are very good too!

Problem # k + 72

Player A has one more coin than player B. Both players throw all of their coins simultaneously and observe the number that come up heads. Assuming all the coins are fair, what is the probability that A obtains more heads than B?

Jai Ganesh · 2005-12-23 16:59:51

John E. Franklin wrote:

ganesh, do you have a list of the problems
that are unsolved? That way I can go back
easier and solve some old ones.

The unsolved problems are.....

# k + 1 - Page 5
# k + 28 - Page 8
# k + 31 - Page 8
# k + 38 - Page 9
# k + 40 - Page 9
# k + 42 - Page 9
# k + 47 - Page 10
# k + 48 - Page 10
# k + 59 - Page 11
# k + 72 - Page 13

krassi_holmz · 2005-12-23 18:05:48

And for k+47
I think I got it. But I'm not so sure.
The sum of all the ten numbers must be "random" number too, because if remainders by 10 are ramdon we can't say anyting about the remainder of the sum. I think that the probability is 0.1

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