Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #76 2009-01-11 23:05:39

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

### Re: 5 Star Questions

ganesh wrote:

The idea is to prove that
is divisible by 30, hence, if
isn't divisible by 5, n would be.

You got it. What you want to prove is that 5 divides
, not
.

And yes, to show that an integer is divisible by 30, it suffices to show that it is divisible by 2, 3 and 5. Since 2, 3 and 5 are all prime, a number that is divisible by all of them will be divisible by their product.

Fermats little theorem states that if

is a prime and
is an integer coprime with
(in other words
does not divide
) then

Offline

## #77 2009-01-11 23:47:02

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

Offline

## #78 2009-01-12 01:26:12

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answer to #36:-
You're right, JaneFairfax!

A very special 'Thanks' for telling me that my proof is acceptable.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #79 2009-01-12 02:46:06

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#37. What is the value of

if
?

#38.Find the number of prime factors in

.

#39. What is the value of

?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #80 2009-01-12 14:23:19

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

### Re: 5 Star Questions

By the way

JaneFairfax wrote:

Since 2, 3 and 5 are all prime, a number that is divisible by all of them will be divisible by their product.

In general, the factors dont have to be prime; they only need to be relatively prime. For example, to prove that a number is divisible by 24, it suffices to show that it is divisible by both 3 and 8. Although 8 is not a prime, we have gcd(3,8) = 1, and thats good enough. Any number that is divisible by 3 and 8 will be divisible by 24.

If the factors are not coprime, then you cant do it this way. For example, 12 is divisible by 4 and 6, but 12 is not divisible by 4 × 6 = 24. 4 and 6 are not coprime  hence you cannot prove that a number is divisible by 24 by proving that it is divisible by 4 and 6,

Offline

## #81 2009-01-12 14:28:18

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

Offline

## #82 2009-01-12 17:35:51

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answer to #37:-
You made a mistake somewhere, JaneFairfax!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #83 2009-01-13 01:03:35

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

### Re: 5 Star Questions

I know what happened. I used
instead of
.

Offline

## #84 2009-01-13 01:23:00

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answer to #37:-
This time correct, JaneFairfax!
Well done!!!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #85 2009-01-13 02:47:56

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#40. What is the value of

?

#41. Find the value of a+b if

.

#42. If

what is the value of
?

#43. Find the value of

(1)

(2)

(3)

(4)

#44. Which is the largest four digit number which is a perfect cube?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #86 2009-01-13 03:07:21

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#45. What is the value of

(1)

?

(2)

?

(3)

?

(4)

?

#46. What is the value of

(1)

?

(2)

?

(3)

?

(4)

?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #87 2009-01-13 07:24:17

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

Offline

## #88 2009-01-13 18:17:10

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answers to #40, #43(1), #43(2), #43(3), #45(2), #45(3), #45(4), #46(1), #46(2), #46(3), and #46(4):-
All the answers are correct, JaneFairfax!
I hope, for all the questions other than #40, you did the calculations in your head and all of them within 10 seconds!
I hope, I believe you solved them that way!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #89 2009-01-14 01:15:44

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#47. Find the least fraction which when added to the expression given below would result in a whole number.

.

#48. A postman goes from a place P to another place Q, his speeds during the first one-third, second one-third and the third one-third of the distance being 4, 5 and 4½ kilometers per hour rsepectively. He returns from Q to P at the speed of 4 kilometers per hour. Find his average speed throughout the journey.

#49. If S = 1³  + 2³  + 3³  + 4³  + ...... n³ , what is the minimum value of n such that S>5000?

#50. A train after travelling 50 kilometers meets with an accident and then travels at

of its former speed and arrives at the destination 35 minutes late.  Had the accident occured 24 kilometers after it did, the train would have reached the destination only 23 minutes late. Find the speed of the train and the distance.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #90 2009-01-14 14:05:52

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

Offline

## #91 2009-01-14 14:06:50

1q2w3e4
Member
Registered: 2009-01-11
Posts: 10

#47
15/16

Offline

## #92 2009-01-14 19:02:38

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answers to #47, #48, #49, and #50:-

1q2w3e4, Your answer to  #47 is correct!
JaneFairfax, answers #48 and #50 are correct.

1q2w3e4 and JaneFairfax,

Very good start, 1q2w3e4!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #93 2009-01-15 00:06:12

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#51. If a+b : b+c : c+a = 6 : 7 : 8 and a + b + c = 14, what is the value of c?

#52. Of two numbers, four times the first is equal to six times the second and the sum of 3 times the first and 6 times the second is 105. Find the first number.

#53. The Greatest Common Divisor and the Least Common Multiple of two numbers are 13 and 455 respectively. If one of the numbers lies between 75 and 125, find the number.

#54. The artio of David's age and Daniel's age is 5:9. The sum of their ages is 56 years. What would be the ratio of their ages after seven years?

#55. In an office, the monthly salary of clerks and officers are in the ratio 3:5. Each clerk contributes 2% of his salary and each officer contributes 3% of his salary to a charitable organisation. If each officer's contribution is \$420, what would be the salary of a clerk?

#56. A circle of area

is inscribed in a square. By how much is the area of the square greater than that of the circle?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #94 2009-01-15 00:31:17

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

Offline

## #95 2009-01-15 00:42:50

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answer to #49:
JaneFairfax,
Please read the question carefully.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #96 2009-01-15 01:14:33

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

Offline

## #97 2009-01-15 01:47:38

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answers to #49, #51, #52, #53, and #54:-

All correct, JaneFairfax!

PS:- Regarding #49, I thought it was easier using the cube sign on the page, rather than LaTeX. I am sorry for the inconvenience caused. My vision is being affected too!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #98 2009-01-16 03:33:07

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

#57. If tan (A+B) = p and tan (A-B) = q, find tan 2B in terms of p and q.

#58. Which of the following statements are true?
(I) Any two lines which are parallel to a third line are also parallel to each other.

(II) Any two planes which are parallel to a third plane are parallel to each other.

(III) Any two lines which are parallel to the same plane are parallel to each other.

(a) I only  (b) II only  (c) I and II only  (d) II and III only  (e) I, II, and III.

#59. The coordinates of two ponts A and B are (2,0) and (0,2) in the XOY plane. If points C and D lie in the first qadrant and ABCD is a square, what is its area?

#60. In the right angled triangle  BCE, angle BCE = 90° and angles CBE and BEC are both 45°. The area of the triangle is 8 square units. With BC as one of it sides, a square ABCD is drawn. What is the area of the square?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## #99 2009-01-16 08:53:55

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

Offline

## #100 2009-01-17 04:15:26

ganesh
Administrator
Registered: 2005-06-28
Posts: 23,359

### Re: 5 Star Questions

Answers to #57, #58, #359, and #60:-

All the answers are right, JaneFairfax!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

## Board footer

Powered by FluxBB