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Math: Number Theory

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Re: Math: Number Theory [#permalink] New post 09 Feb 2014, 08:07
any body can explain that why we need to find the sum of number's factors ????
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Re: Math: Number Theory [#permalink] New post 01 May 2014, 06:50
Thanks for the post!! So helpful!
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Re: Math: Number Theory [#permalink] New post 21 Jun 2014, 03:08
Hi Bunuel,
You can change this section to.

Verifying the primality (checking whether the number is a prime) of a given number n can be done by trial division, that is to say dividing n by all integer prime numbers smaller than \sqrt{n}, thereby checking whether n is a multiple of m<\sqrt{n}.
Example: Verifying the primality of 161: \sqrt{161} is little less than 13, from integers from 2 to 13, 161 is divisible by 7, hence 161 is not prime.
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Re: Math: Number Theory [#permalink] New post 24 Jul 2014, 15:03
new to here. precious compilation. Great job
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Re: Math: Number Theory [#permalink] New post 14 Aug 2014, 16:17
Hi -

Can somebody explain where the '00' part of the 9900 in the denominator is coming from? I understand where the '99' is coming from. I copied this from the number theory section.

Thanks

" Example #2: Convert 0.2512(12) to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300. "
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Re: Math: Number Theory [#permalink] New post 14 Aug 2014, 19:25
Bunuel,

I am confused regarding the following exponent rule in the Math Book:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\)

Isn't \(a^m^n=a^{(m*n)}\)?

Thanks in advance!
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Re: Math: Number Theory [#permalink] New post 15 Aug 2014, 00:44
Expert's post
prudhviram wrote:
Bunuel,

I am confused regarding the following exponent rule in the Math Book:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\)

Isn't \(a^m^n=a^{(m*n)}\)?

Thanks in advance!


No.

\((a^m)^n=a^{mn}\)

\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).
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Re: Math: Number Theory [#permalink] New post 15 Aug 2014, 03:06
Bunuel wrote:
prudhviram wrote:
Bunuel,

I am confused regarding the following exponent rule in the Math Book:
\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\)

Isn't \(a^m^n=a^{(m*n)}\)?

Thanks in advance!


No.

\((a^m)^n=a^{mn}\)

\(a^m^n=a^{(m^n)}\) and not \((a^m)^n\) (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).


I guess when the exponentiation is stacked, its not much of a confusion.

I was tripped by the following problem:
If n = 10^10 and (n^n) = (10^d), what is the value of d? (I know there is a different thread for this problem, but I wanted to refer it here for the sake of clarity)

Here the exponentiation 10^10^10^10 is not explicitly stacked. How do I approach such problems?
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Re: Math: Number Theory [#permalink] New post 17 Sep 2014, 21:09
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Hi -

Can somebody explain where the '00' part of the 9900 in the denominator is coming from? I understand where the '99' is coming from. I copied this from the number theory section.

Thanks

" Example #2: Convert 0.2512(12) to a fraction.
1. The number consisting with non-repeating digits and repeating digits is 2512;
2. Subtract 25 (non-repeating number) from above: 2512-25=2487;
3. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900=829/3300. "


let x =0.2512(12)
100x = 25.(12)
10000x = 2512.(12)

10000x -100x = 2512.(12) - 25.(12) = 2487
9900x = 2487
x = 2487/9900

That is the logic. Hope that helps.
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Re: Math: Number Theory [#permalink] New post 20 Sep 2014, 07:44
Guys can anyone explain this line in factorial factorization?

"We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!."

Why do the primes have to be represented twice?
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Re: Math: Number Theory [#permalink] New post 20 Sep 2014, 12:31
Expert's post
converge wrote:
Guys can anyone explain this line in factorial factorization?

"We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!."

Why do the primes have to be represented twice?


Because 900 = 2^2*3^2*5^2 = 2*2*3*3*5*5.
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Re: Math: Number Theory [#permalink] New post 03 Nov 2014, 11:02
As a part of an HR initiative, each day the MD of Telair Ltd. decides to have breakfast with 4 other employees, such that each day there is a different set of 4 employees. If he has 15 employees in all, how many ways MD can have breakfast with the employees?
please help me with this!!
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Re: Math: Number Theory [#permalink] New post 03 Nov 2014, 11:03
As a part of an HR initiative, each day the MD of Telair Ltd. decides to have breakfast with 4 other employees, such that each day there is a different set of 4 employees. If he has 15 employees in all, how many ways MD can have breakfast with the employees?
please help me with this!!
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Re: Math: Number Theory [#permalink] New post 03 Nov 2014, 11:04
In how many ways can a committee of 10 be selected with at least 4 women and atleast 4 men from 7 women and 8 men if Ms X refuses to work with Mr. Y. How to approach such questions ?? please help.
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Re: Math: Number Theory [#permalink] New post 04 Nov 2014, 04:11
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In how many ways can a committee of 10 be selected with at least 4 women and atleast 4 men from 7 women and 8 men if Ms X refuses to work with Mr. Y. How to approach such questions ?? please help.


Please post question in relevant forum. How to post: rules-for-posting-please-read-this-before-posting-133935.html
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Re: Math: Number Theory [#permalink] New post 14 Nov 2014, 15:57
:-D this is great
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Re: Math: Number Theory [#permalink] New post 06 Dec 2014, 08:23
Bunuel, Can't we also add a small point to this theory, viz. that NO Perfect Square end with the unit digits as 2,3,7,8.
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Re: Math: Number Theory [#permalink] New post 14 Jan 2015, 17:38
Bunuel wrote:
defoue wrote:
Hi buddies,
going through this awesome contribution, I got some issues to understand the following : would you please help to understand by giving some extra examples?

1. Finding the power of non-prime in n!:

How many powers of 900 are in 50!

Make the prime factorization of the number: 900=2^2*3^2*5^2, then find the powers of these prime numbers in the n!.

Find the power of 2:
\frac{50}{2}+\frac{50}{4}+\frac{50}{8}+\frac{50}{16}+\frac{50}{32}=25+12+6+3+1=47

= 2^{47}

Find the power of 3:
\frac{50}{3}+\frac{50}{9}+\frac{50}{27}=16+5+1=22

=3^{22}

Find the power of 5:
\frac{50}{5}+\frac{50}{25}=10+2=12

=5^{12}

We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.

2. I did not get the tips 3 or 4 regarding the pefect square whare it says a perfect square has an odd nbr of odd powers and an even nbr of even power
what aboute 36 which is 2^{2} x 3^{2}
I am sure I am missing something here..


Thx guys


1. It's highly unlikely that this concept will be tested in GMAT. But still:

Suppose we have the number \(18!\) and we are asked to to determine the power of \(12\) in this number. Which means to determine the highest value of \(x\) in \(18!=12^x*a\), where \(a\) is the product of other multiples of \(18!\).

\(12=2^2*3\), so we should calculate how many 2-s and 3-s are in \(18!\).

Calculating 2-s: \(\frac{18}{2}+\frac{18}{2^2}+\frac{18}{2^3}+\frac{18}{2^4}=9+4+2+1=16\). So the power of \(2\) (the highest power) in prime factorization of \(18!\) is \(16\).

Calculating 3-s: \(\frac{18}{3}+\frac{18}{3^2}=6+2=8\). So the power of \(3\) (the highest power) in prime factorization of \(18!\) is \(8\).

Now as \(12=2^2*3\) we need twice as many 2-s as 3-s. \(18!=2^{16}*3^8*a=(2^2)^8*3^8*a=(2^2*3)^8*a=12^8*a\). So \(18!=12^8*a\) --> \(x=8\).




I am still struggling with this somewhat. Could you by chance off another practice problem with explanation? Or explain the problem from the guide? I don't understand how we determine that 5 can only provide us with 6 pairs when we have 5^12
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Re: Math: Number Theory [#permalink] New post 15 Jan 2015, 00:08
Expert's post
mmallison wrote:
Bunuel wrote:
defoue wrote:
Hi buddies,
going through this awesome contribution, I got some issues to understand the following : would you please help to understand by giving some extra examples?

1. Finding the power of non-prime in n!:

How many powers of 900 are in 50!

Make the prime factorization of the number: 900=2^2*3^2*5^2, then find the powers of these prime numbers in the n!.

Find the power of 2:
\frac{50}{2}+\frac{50}{4}+\frac{50}{8}+\frac{50}{16}+\frac{50}{32}=25+12+6+3+1=47

= 2^{47}

Find the power of 3:
\frac{50}{3}+\frac{50}{9}+\frac{50}{27}=16+5+1=22

=3^{22}

Find the power of 5:
\frac{50}{5}+\frac{50}{25}=10+2=12

=5^{12}

We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.

2. I did not get the tips 3 or 4 regarding the pefect square whare it says a perfect square has an odd nbr of odd powers and an even nbr of even power
what aboute 36 which is 2^{2} x 3^{2}
I am sure I am missing something here..


Thx guys


1. It's highly unlikely that this concept will be tested in GMAT. But still:

Suppose we have the number \(18!\) and we are asked to to determine the power of \(12\) in this number. Which means to determine the highest value of \(x\) in \(18!=12^x*a\), where \(a\) is the product of other multiples of \(18!\).

\(12=2^2*3\), so we should calculate how many 2-s and 3-s are in \(18!\).

Calculating 2-s: \(\frac{18}{2}+\frac{18}{2^2}+\frac{18}{2^3}+\frac{18}{2^4}=9+4+2+1=16\). So the power of \(2\) (the highest power) in prime factorization of \(18!\) is \(16\).

Calculating 3-s: \(\frac{18}{3}+\frac{18}{3^2}=6+2=8\). So the power of \(3\) (the highest power) in prime factorization of \(18!\) is \(8\).

Now as \(12=2^2*3\) we need twice as many 2-s as 3-s. \(18!=2^{16}*3^8*a=(2^2)^8*3^8*a=(2^2*3)^8*a=12^8*a\). So \(18!=12^8*a\) --> \(x=8\).




I am still struggling with this somewhat. Could you by chance off another practice problem with explanation? Or explain the problem from the guide? I don't understand how we determine that 5 can only provide us with 6 pairs when we have 5^12


For similiar questions check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.



Hope this helps.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Math: Number Theory [#permalink] New post 19 Jan 2015, 21:10
Are there practice problems related to number there in the OG for 12th edition or any edition after that? I see the practice problems after the percents, but those all relate to percent questions.

Anything questions in the OGs that anyone knows of related to number theory and these rules would be great practice. Thanks!
Re: Math: Number Theory   [#permalink] 19 Jan 2015, 21:10

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