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**kylekatarn****Member**- Registered: 2005-07-24
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ganesh wrote:

Excellent, Toast and Devante`! Well done! Don't mind if it took a second or two more!

84. How many digits does 100! contain?

85. And how many digits does 2^500 contain?

84) 158

85) 151

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**ganesh****Moderator**- Registered: 2005-06-28
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Excellent, kylekatarn!

Welcome back to the forum!

Character is who you are when no one is looking.

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**Toast****Real Member**- Registered: 2006-10-08
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mathsyperson wrote:

Depending on where you are, the answer to 80. could also be 54 zeroes.

85. 500/log 2, rounded up.

Wow... that's amazing! Is there a reason why logging it will give you the number of digits?

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Yep. I made a mistake there, it should actually be 500*log 2, rounded up. But with the correction, it works.

You can see more easily why it works if we were asked how many digits were in 10^500. There are 501 digits in 500, just like there are 3 in 100 = 10², or 2 in 10 or whatever.

By definition, log (10^500) = 500, log 10² = 2, etc.

The reason that the results are all off by one is because they are all integers, so we can't round them up because they're already rounded.

Anyway, the laws of logs say that to take a different base of log of the number, you just need to take the log of the number and divide the result by the log of the base.

Rearranging that equation gives us the one that I used: {number of digits} = 500*log 2.

Why did the vector cross the road?

It wanted to be normal.

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**kylekatarn****Member**- Registered: 2005-07-24
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ganesh wrote:

Excellent, kylekatarn!

Welcome back to the forum!

Thanks, its good to be back : )

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**ganesh****Moderator**- Registered: 2005-06-28
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86. How many distinct prime factors does 2400 have?

87. And how many distinct prime factors does 8400 have?

88. February 5, 2007 is a Monday. What day would February 5, 2008 be?

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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86. Three: 2, 3, 5.

87. Four: 2, 3, 5, 7.

88. Tuesday.

Why did the vector cross the road?

It wanted to be normal.

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**kylekatarn****Member**- Registered: 2005-07-24
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*EDIT*: mathsy got it a few minutes faster:P

*Last edited by kylekatarn (2007-02-05 01:44:37)*

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**ganesh****Moderator**- Registered: 2005-06-28
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Excellent, mathsy and kylekatarn!

89. √2 + √3 + √4 + √5 is closest to which whole number?

90. What is the difference between (9x9x9x9) and (8x8x8x8)?

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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89. (1 and a bit) + (1 and a big bit) + 2 + (2 and a little bit) = ~7?

90. (8+1)^4 = 8^4 + 4*8^3 + 6*8^2 + 4*8 + 1

∴ 9^4 - 8^4 = 4*512 + 6*64 + 32 + 1 = 2465.

Bit of a strange method, but that's the one that I think would be fastest for me.

Why did the vector cross the road?

It wanted to be normal.

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**Devantè****Real Member**- Registered: 2006-07-14
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89. √2 + √3 + √4 + √5 ≈ 7

90. 6561-4096 = 2465

It really helps if you know your cube numbers. I just did (7³×7)-(8³×8). In other words, (729x7)-(512x8). Quick arithmetic, 6561-4096 = 2465.

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**ganesh****Moderator**- Registered: 2005-06-28
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Well done, mathsyperson and Devante`.

I think I'd solve the second problem, viz. 90 the way Devante' did. Thats because I can tell from memory values of 9^4 and 8^4.

91. 3 feet is a yard, 220 yards a furlong and 8 furlongs a mile. How many feet is a mile?

92. Which of the two is greater? 100xpi or √100000?

93. How many numbers are there from 1 to 1000 which are divisible by 2, 3 and 5 but not divisible by 7?

Character is who you are when no one is looking.

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**Maelwys****Member**- Registered: 2007-02-02
- Posts: 161

Answers:

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Welcome to the forum, Maelwys!

Very impressive working out for 93. I think you might have made one small error though.

Incidentally, I read somewhere ages ago that the number of feet in a mile is closely approximated by

.Completely useless, but still fun.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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I agree with mathsyperson. But I shouldn't have posted this question in the 10 seconds section. I am sure that would never have been enough.

Well done, mathsyperson!

Excellent, Maelwys! Welcome to the forum!!!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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94. Can you think of a triangle whose angles are in Arithmetic Progression?

95. A triangle cannot have angles (in whole numbers in degrees) in Gemoetric Progression. True of False?

Character is who you are when no one is looking.

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**Toast****Real Member**- Registered: 2006-10-08
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94. 30, 60, and 90 degrees ^^

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**mathsyperson****Moderator**- Registered: 2005-06-22
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There are actually lots of triangles that would work for 94. Equilaterals are another example. The only condition you need is that one of the angles is 60[sup]o[/sup], if that's true then the other angles will always fit.

95 is trickier. I'll have a think about that one.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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Yes, both Toast and mathsyperson are correct! Good work!

96. Irrespective of whether a year is a leap year or not, the months September and December have the same days for the same dates. Can you think of another such pair?

97. If a tyre 1 metre in daimeter runs at 60 revolutions per minute for an hour, what is the approximate distance covered?

98. How long would a train 200 metres long running at 10 metres/second take to cross a bridge 600 metres long?

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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99. If Sn denotes the sum of n terms of an Arithmetic Progression and S1:S4 = 1:10, then what is the ratio of the first term to the fourth term?

100. The lengths of a triangle are x+1, 9-x and 5x-3. For how many values of x is the triangle isosceles?

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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Excellent, mathsyperson!!!!!

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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101. (330x14)/5=?

102. 41.2 x 0.15 = ?

103. If 24 men can do a work in 40 days, in how many days would 30 men be able to do the same work?

104. A motorist tarvels at a constant speed of 82.6 Kilometers/hour. How many meters would he travel in 15 minutes?

105. $6200 amounts to $9176 in 4 years at simple interest. What is the rate of interest?

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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106. Find the quadratic equation that has the roots 4 and -4.

107. What is the 16th term of the Arithmetic Progression 1, 6, 11, 16, 21......

108. If (5x³ - 2x + a) is divided by (x-2), the remainder is 7. What is the value of a?

Character is who you are when no one is looking.

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**kylekatarn****Member**- Registered: 2005-07-24
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*Last edited by kylekatarn (2007-02-11 01:50:44)*

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