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#1 2006-02-28 04:13:03

ganesh
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Registered: 2005-06-28
Posts: 15,169

Number Theory

NT # 1

What is the highest power of 7 in 5000!?


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#2 2006-02-28 04:21:32

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Number Theory

5000! = 5000*4999*4998*4997*...*3*2*1.

Of all the numbers up to 5000, there are 5000/7 = 714 numbers that have 7 as a factor.
Of these 714, 714/7 = 102 of them have 7² as a factor.
Of these 102, 102/7 = 14 of them have 7³ as a factor.
Of these 14, 14/7 = 2 of them have 7[sup]4[/sup] as a factor.

Adding all of these gives 832, so that is the highest power of 7 in 5000!.


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It wanted to be normal.

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#3 2006-02-28 04:28:26

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

TE10B_B_GoodWorkApple.jpg


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#4 2006-02-28 06:02:06

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

There's a formula:
The higest power of p(prime) that divides n! is exactly:

Last edited by krassi_holmz (2006-02-28 06:02:59)


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#5 2006-02-28 20:18:25

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

NT # 2

If both 11²  and 3³  are factors of the number a * 4³ * 6² *13[sup]11[/sup], then what is the smallest possible value of a?


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#6 2006-02-28 20:23:06

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

6^2=3^2 2^2
a=3*11^2


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#7 2006-02-28 23:49:56

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

krassi_holmz, you get this award!

job-well-done.jpg


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#8 2006-03-01 00:00:53

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

NT # 3

A person starts multiplying consecutive positive integers from 20. How many numbers should he multiply before he will have a result that will end with 3 zeroes?


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#9 2006-03-01 02:24:09

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

20*21*23*24*25=127512000.


IPBLE:  Increasing Performance By Lowering Expectations.

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#10 2006-03-01 03:36:47

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

Correct! cool


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#11 2006-03-01 20:48:23

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

NT # 4

What is the Least Common Multiple of 3/8, 5/36, 7/72 and 15/96?


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#12 2006-03-02 02:25:02

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

105/4


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#13 2006-03-02 02:48:20

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

try.jpg


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#14 2006-03-02 03:15:37

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

What?
105/4=3/8*70=3/8*(2*5*7)
105/4=5/36*189=5/36*(3*3*3*7)
105/4=7/72*270=7/72*(2*5*3*3*3)
105/4=15/96*168=15/96*(7*3*2*2*2)
There ISN"T a number LESS than 105/4


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#15 2006-03-02 03:36:49

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

krassi_holmz,

What about 105/288?


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#16 2006-03-02 17:53:35

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

NT # 5

In an examination, the average marks obtained by students who passed was x%, the average marks obtained by students who failed was y% and the average marks of all the students who appeared for the examintion was z%. Find the percentage of students who failed in the examination in terms of x, y, and z.


Character is who you are when no one is looking.

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#17 2006-03-03 00:06:29

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: Number Theory

for NT 4:
(105/288):(3/8)=(35/36), which is not an integer???


IPBLE:  Increasing Performance By Lowering Expectations.

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#18 2006-03-03 02:29:58

ganesh
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Registered: 2005-06-28
Posts: 15,169

Re: Number Theory

I concede, you are correct krassi_holmz, well done! smile


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#19 2008-06-02 05:19:13

dan howitt
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Registered: 2008-05-14
Posts: 21

Re: Number Theory

ganesh wrote:

Nice!


Dan Howitt

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#20 2008-07-21 14:35:10

Sudeep
Member
Registered: 2008-07-21
Posts: 20

Re: Number Theory

Want help to solve few questions...simple for people having knowledge of Euler/ Fermet theorem but i am not that great at it...

1) Two numbers 698 and 450 when divided by a certain divisor leave remainders of 9 and 8 respectively. Find the largest such divisor.

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#21 2008-07-21 14:59:00

ZHero
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Registered: 2008-06-08
Posts: 1,889

Re: Number Theory

If upon dividing 698 and 450, the divisor leaves remainders 9 and 8 respectively then it Must divide (698-9=689) and (450-8=442)!
Therefore, the required number is HCF of (689=13*53) and (442=2*13*17) which clearly is 13 !
Hence 13 is the answer...


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#22 2008-07-21 21:37:22

Sudeep
Member
Registered: 2008-07-21
Posts: 20

Re: Number Theory

What is the minimum number of identical square tiles required to completely cover a floor of dimensions 8m 70 cm by 6m 38cm

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#23 2008-07-21 23:05:43

ZHero
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Registered: 2008-06-08
Posts: 1,889

Re: Number Theory

Hi BumBumBhole!
From India haan?
Well.. Me too!
Jai Shiv Shankar!

Coming to the point..
Minimum number of squares will be needed if their size is maximum!
So required size of the squares is the square with sides of length which is HCF of 870cm & 638cm i.e. 29cm !
And number of squares is area of floor divided by area of tile..
=(870*638)/(29*29)
=330

Easy! Ain't it?

Last edited by ZHero (2008-07-21 23:30:05)


If two or more thoughts intersect with each other, then there has to be a point.

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#24 2008-07-22 00:52:38

Sudeep
Member
Registered: 2008-07-21
Posts: 20

Re: Number Theory

ok learnt the approach the HCF will be 58 and not 29 so the answer will be 870*638/58^2 = 165

thnx anyway

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#25 2008-07-22 01:00:34

Sudeep
Member
Registered: 2008-07-21
Posts: 20

Re: Number Theory

Another one ...the total number of factors of a natural number N is 45. What is the maximum number of prime numbers by which N can be divided?

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