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Math: Number Theory

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27 Feb 2011, 10:04
Any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

Pls give me the example of bold face text because i am not sure what does it exactly means.

Thanks
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27 Feb 2011, 10:12
GMATD11 wrote:
Any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

Pls give me the example of bold face text because i am not sure what does it exactly means.

Thanks

It's called the fundamental theorem of arithmetic (or the unique-prime-factorization theorem) which states that any integer greater than 1 can be written as a unique product of prime numbers.

For example: 60=2^2*3*5 --> 60 can be written as a product of primes (powers of primes) only in this unique way (you can just reorder the multiples and write 3*2^2*5 or 2^2*5*3 ...).
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27 Feb 2011, 11:55
If a is a factor of bc, and gcd(a,b)=1, then a is a factor of c.

any particular example.
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03 Mar 2011, 05:14
Nice work mates....very informative source to kickkkk start my Prep.....
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06 Mar 2011, 05:50
Can someone please explain the following:

1. "Special Cases" section in "Evenly Spaced Integers"

2. Last digit of a power

Thanks.
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27 May 2011, 08:21
thank you for the great post. I currently use the GMAT Toolkit app, which I highly recommend, when can I expect this update? In addition, when will the Manhattan GMAT books be updated to the app?

Thanks,
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27 May 2011, 09:24
Hi,

Thanks for your words! 1.6.0 update is available for download. Just get it, go to Store and you can buy any of 10 famous Manhattan GMAT books.

Let me know if you have any questions.

OrenY wrote:
thank you for the great post. I currently use the GMAT Toolkit app, which I highly recommend, when can I expect this update? In addition, when will the Manhattan GMAT books be updated to the app?

Thanks,

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17 Jul 2011, 10:18
is this always true?
The product of n consecutive integers is always divisible by n!.
Given consecutive integers: . The product of 3*4*5*6 is 360, which is divisible by 4!=24
.

for example, n=10 and the first number starts at 100000, then this rule doesn't hold.
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05 Sep 2011, 01:15
Hi GMAT CLUB,

Thank you for this super book.

I am completely following it for my exam in November.

I am going through the chapter on Number Theory, under heading "Finding the Number of Factors of an Integer".

Please can someone explain how you arrived at the following rule. Is there any proof for it.

For an integer n=a^p*b^q*c^r, where a, b, and c are prime factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1).

NOTE: this will include 1 and n itself.

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10 Sep 2011, 23:04
This one piece is awesome of all on the math book!
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10 Sep 2011, 23:08
gujralam wrote:
Hi GMAT CLUB,

Thank you for this super book.

I am completely following it for my exam in November.

I am going through the chapter on Number Theory, under heading "Finding the Number of Factors of an Integer".

Please can someone explain how you arrived at the following rule. Is there any proof for it.

For an integer n=a^p*b^q*c^r, where a, b, and c are prime factors of n and p, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1).

NOTE: this will include 1 and n itself.

Please search for "unique-prime-factorization theorem" on web, you should be able to get what you are looking for!
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21 Sep 2011, 02:37
Thanks a lot Bunuel.. truly awesome...
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24 Nov 2011, 18:23
This is THE BEST thing anyone has ever posted. THANK YOU SO MUCH.

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17 Dec 2011, 08:52
Hi, thanks for the great summary. BTW, do you have a list of questions (just question number) in OG12 + Quant Review 2nd edition to practice, just like the Triangles and Circle section?

Thanks again!
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06 Feb 2012, 15:55
Bunnel you simply ROCK!!!!
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05 Mar 2012, 01:29
Breathtaking post! (Literally!)
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06 Mar 2012, 13:00
Bunuel wrote:
NUMBER THEORY

EXPONENTS

Exponents and divisibility:
$$a^n-b^n$$ is ALWAYS divisible by $$a-b$$.
$$a^n-b^n$$ is divisible by $$a+b$$ if $$n$$ is even.
$$a^n + b^n$$ is divisible by $$a+b$$ if $$n$$ is odd, and not divisible by a+b if n is even.

Hello, Bunuel. Great post!

Do you have an example problem in which this applies. I plugged in numbers to understand the concept I was just curious about the application and seeing this in action. Thanks.
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06 Mar 2012, 13:03
destroyerofgmat wrote:
Bunuel wrote:
NUMBER THEORY

EXPONENTS

Exponents and divisibility:
$$a^n-b^n$$ is ALWAYS divisible by $$a-b$$.
$$a^n-b^n$$ is divisible by $$a+b$$ if $$n$$ is even.
$$a^n + b^n$$ is divisible by $$a+b$$ if $$n$$ is odd, and not divisible by a+b if n is even.

Hello, Bunuel. Great post!

Do you have an example problem in which this applies. I plugged in numbers to understand the concept I was just curious about the application and seeing this in action. Thanks.

Check this: if-n-is-an-integer-1-is-3-n-2-n-divisible-by-84992.html
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06 Mar 2012, 14:32
Bunuel wrote:
destroyerofgmat wrote:
Bunuel wrote:
NUMBER THEORY

EXPONENTS

Exponents and divisibility:
$$a^n-b^n$$ is ALWAYS divisible by $$a-b$$.
$$a^n-b^n$$ is divisible by $$a+b$$ if $$n$$ is even.
$$a^n + b^n$$ is divisible by $$a+b$$ if $$n$$ is odd, and not divisible by a+b if n is even.

Hello, Bunuel. Great post!

Do you have an example problem in which this applies. I plugged in numbers to understand the concept I was just curious about the application and seeing this in action. Thanks.

Check this:

Awesome! Thanks. That's definitely above my level but good practice no doubt.
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17 May 2012, 02:03
Thanks a Tonn for the detailed post and help !!!!!
Re: Math: Number Theory   [#permalink] 17 May 2012, 02:03

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