Problems and Solutions

Jai Ganesh · 2005-10-11 20:22:13

Problem # k + 28

For any value of n (n∈N), prove that n²! is divisible by (n!)^n.

Jai Ganesh · 2005-10-13 16:58:27

Problem # k + 29

If each interior angle of a regular polygon is 150 degrees, how many sides does it contain?

justlookingforthemoment · 2005-10-13 20:27:58

For #k +29

Hide tags don't like degree symbols.

Last edited by justlookingforthemoment (2005-10-13 20:32:03)

Jai Ganesh · 2005-10-13 20:44:46

Excellent! I used a

method for finding the solution.
Well done, you have made a great start!

justlookingforthemoment · 2005-10-13 20:51:00

Thank you, ganesh!

Jai Ganesh · 2005-10-14 16:32:04

Problem # k + 30

If a sum of money grows to 144/121 times when invested for two years in a scheme where interest is compounded annually, how long will the same sum of money take to treble if invested at the same rate of interest in a scheme where interest is computed using simple interest method?

mathsyperson · 2005-10-14 22:29:46

I can do k+28 to my own satisfaction, but it's really hard to put it into words, so I won't.

k+1 and k+28 remain.

Last edited by mathsyperson (2005-10-14 22:29:59)

Jai Ganesh · 2005-10-15 00:01:38

You are right, Mathsy.
For k + 28, you may not require too many words, just mathematical expressions. I know you can do it. To attempt or not is your decision.I shall post the solution soon.

Jai Ganesh · 2005-10-15 17:13:49

Problem # k + 31

From a lot of 10 items, of which three are defective, a sample of four is drawn. Which of these is most likely and what is the probability?
(a) One defective item is drawn
(b) Two defective items are drawn
(c) Three defective items are drawn
(d) No defective item is drawn

Jai Ganesh · 2005-10-15 17:20:19

Problem # k + 32

Show that the three cube roots of 1 are in Geometric Progression.

MathsIsFun · 2005-10-15 17:43:06

k+32
Gee, isn't it enough just to FIND the cube roots of 1?

Let me see, the square roots of 1 would be: 1 and -1, right?

And the cube roots of 1 are: 1 and ...

Oh, I could look it up but it is some combination of a real and imaginary, because it can't be real (because we have already used 1, and -1 doesn't work), and it can't be all imaginary because then the cube would end up imaginary

So they are going to be of the form (a+bi)

The cube of (a+bi) is: a³ + 3a²bi + 3a(bi)² + (bi)³

And it must end up as 1

mathsyperson · 2005-10-15 21:40:12

a³ + 3a²bi + 3a(bi)² + (bi)³

Using i² = -1, that can become a³ + 3a²bi - 3ab² - b³i

Split up the real and imaginary terms: a³ - 3ab² = 1 ; 3a²b - b³ = 0

And there you have two rather nasty simulataneous equations that should give answers of the form a ± bi.
But they're horrible.

MathsIsFun · 2005-10-15 21:45:40

I found the answer on another website, but no explanation of how it was arrived at, so I think it would be more fun to figure it out. And we seem to be on a good path.

Jai Ganesh · 2005-10-16 16:11:36

Solution to Problem # k + 32

x³ = 1
x³ - 1 = 0
(x - 1)(x²+x+1)=0
x = 1 or (x²+x+1)=0
Solving the second equation, we get
x = (-1 ± √3i)/2
Hence, the cube roots of 1 are
1, (-1 - √3i)/2, (-1 + √3i)/2

It can be shown that the three are in Geometic progression with common ratio r= (-1 - √3i)/2
When (-1 - √3i)/2 is multiplied by itself, we get (-1 + √3i)/2
quod erat faciendum

Jai Ganesh · 2005-10-21 15:50:14

Problem # k + 33

What are the values of M and N if M39048458N is divisible by 8 & 11 where M & N are single digit whole numbers?

Jai Ganesh · 2005-10-21 15:55:34

Problem # k + 34

Two numbers when divided by a certain divisor leave remainders of 431 and 379 respectively. When the sum of these two numbers is divided by the same divisor, the remainder is 211. What is the divisor?

mathsyperson · 2005-10-21 22:03:29

Jai Ganesh · 2005-10-21 22:52:24

Thats correct, Mathsy!

Jai Ganesh · 2005-10-21 23:09:16

Problem # k + 35

An easy one:-

Dazzy fishes for more than a week. He experiences three different levels of success. On a good day he catches 9 fishes. On a bad day he catches only 5 fishes. On other days he catches 7 fishes.
In all, Dazzy catches 53 fishes. How many bad days did he have?

Jai Ganesh · 2005-10-21 23:39:25

Problem # k + 36

Is it true that n^3+5n-1 is prime for any natural n?

mathsyperson · 2005-10-22 01:28:16

Last edited by mathsyperson (2005-10-22 01:28:35)

Jai Ganesh · 2005-10-22 18:35:03

Jai Ganesh · 2005-10-22 18:39:03

Problem # k + 37

A string of 1996 digits begins with the number 6. Any number formed by two consecutive digits is divisible by 17 or 23. The number contains two consecutive digits which are multiples of 17 and multiples of 23. What is the last digit?

Jai Ganesh · 2005-10-24 16:14:55

Problem # k + 38

Prove that, for any natural number n ≥ 2, there exists atleast one prime number between n and 2n.

Jai Ganesh · 2005-10-25 16:22:16

Problem # k + 39

Of all the five digit numbers, how many are odd and without repetition of digits?

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