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**JaneFairfax****Member**- Registered: 2007-02-23
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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.

Fermats little theorem states that if

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

**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #36:-

You're right, JaneFairfax!

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

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#37. What is the value of

?

#38.Find the number of prime factors in

#39. What is the value of

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**JaneFairfax****Member**- Registered: 2007-02-23
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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 dont 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 thats good enough. Any number that is divisible by 3 and 8 will be divisible by 24.

If the factors are not coprime, then you cant 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,

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #37:-

You made a mistake somewhere, JaneFairfax!

Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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I know what happened. I used instead of .

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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #37:-

This time correct, JaneFairfax!**Well done!!!**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#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?

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**ganesh****Moderator**- Registered: 2005-06-28
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#45. What is the value of

(1)

?(2)

?(3)

?(4)

?#46. What is the value of

(1)

?(2)

?(3)

?(4)

?Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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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!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#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.Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**1q2w3e4****Member**- Registered: 2009-01-11
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#47

15/16

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**ganesh****Moderator**- Registered: 2005-06-28
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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!**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#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?Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answer to #49:

JaneFairfax,

Please read the question carefully.

Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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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!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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#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?

Character is who you are when no one is looking.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**ganesh****Moderator**- Registered: 2005-06-28
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Answers to #57, #58, #359, and #60:-

All the answers are right, JaneFairfax!

Character is who you are when no one is looking.

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