Math: Number Theory : GMAT Quantitative Section - Page 6
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# Math: Number Theory

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27 May 2011, 08:24
Hi,

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17 Jul 2011, 09:18
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, 00:15
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, 22:04
This one piece is awesome of all on the math book!
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10 Sep 2011, 22:08
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, 01:37
Thanks a lot Bunuel.. truly awesome...
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24 Nov 2011, 17:23
This is THE BEST thing anyone has ever posted. THANK YOU SO MUCH.

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17 Dec 2011, 07:52
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, 14:55
Bunnel you simply ROCK!!!!
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05 Mar 2012, 00:29
Breathtaking post! (Literally!)
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06 Mar 2012, 12:00
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, 12:03
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, 13:32
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, 01:03
Thanks a Tonn for the detailed post and help !!!!!
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01 Jun 2012, 12:27
Thanks a lot for the detailed post ! This is sure to help with my recurrent mistakes on number properties
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04 Sep 2012, 18:34
Bunuel wrote:
NUMBER THEORY

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer $$n$$, can be determined with this formula:

$$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that $$5^k<n$$.

It's easier if you look at an example:

How many zeros are in the end (after which no other digits follow) of $$32!$$?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

Hence, there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

I noticed in case the number (n) is multiple of $$5^k$$ and we have to find number of trailing zero zeroes, then it will be $$5^k<=n$$ rather $$5^k<n$$

no of trailing zeros in 25! =6

$$\frac{25}{5}+\frac{25}{5^2}= 5+1$$;
Please correct me, clarify if i'm wrong. Thanks
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04 Sep 2012, 23:13
conty911 wrote:
Bunuel wrote:
NUMBER THEORY

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer $$n$$, can be determined with this formula:

$$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that $$5^k<n$$.

It's easier if you look at an example:

How many zeros are in the end (after which no other digits follow) of $$32!$$?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

Hence, there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

I noticed in case the number (n) is multiple of $$5^k$$ and we have to find number of trailing zero zeroes, then it will be $$5^k<=n$$ rather $$5^k<n$$

no of trailing zeros in 25! =6

$$\frac{25}{5}+\frac{25}{5^2}= 5+1$$;
Please correct me, clarify if i'm wrong. Thanks

You are right. $$k$$ is the highest power of 5 not exceeding $$n.$$
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22 Sep 2012, 19:15
Hey..Awesome post !!
Like number theory, can u pls share links for other topics as well ?
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24 Sep 2012, 03:01
154238 wrote:
Hey..Awesome post !!
Like number theory, can u pls share links for other topics as well ?

Check all topics in our Math Book: gmat-math-book-87417.html

Hope it helps.
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25 Sep 2012, 09:43
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conty911 wrote:
Bunuel wrote:
NUMBER THEORY

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer $$n$$, can be determined with this formula:

$$\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}$$, where k must be chosen such that $$5^k<n$$.

It's easier if you look at an example:

How many zeros are in the end (after which no other digits follow) of $$32!$$?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

Hence, there are 7 zeros in the end of 32!

The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

I noticed in case the number (n) is multiple of $$5^k$$ and we have to find number of trailing zero zeroes, then it will be $$5^k<=n$$ rather $$5^k<n$$

no of trailing zeros in 25! =6

$$\frac{25}{5}+\frac{25}{5^2}= 5+1$$;
Please correct me, clarify if i'm wrong. Thanks

The highest power of a prime number "k" that divides any number "n!" is given by the formula
n/K + n/k^2+n/k^3.. (until numerator becomes lesser than the denominator). Remember to truncate the remainders of each expression

E.g : The highest number of 2's in 10! is
10/2 + 10/4 + 10/8 = 5 + 2 + 1 = 8 (Truncate the reminder of each expression)

As a consequence of this, the number of zeros in n! is controlled by the presence of 5s.
Why ? 2 reasons

a) 10 = 5 x 2,
b) Also in any n!, the number of 5's are far lesser than the number of 2's.

The number of cars that you make depends on the number of engines. You can have 100 engines and 1000 cars, but you can only make 100 cars (each car needs an engine !)

10 ! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Lets factorize each term ...
10! = (5 x 2) x(3x3)x(2x2x2)x7x(2x3)x(5)X(2x2)x1
the number of 5s = 2
The number of 2s = 7
The number of zeros in 10! = the total number of 5s = 2 (You may use a calc to check this10! = 3628800)

hence in any n! , the number of 5's control the number of zeros.

As a consequence of this, the number of 5's in any n! is
n/5 + n/25 + n/125 ..until numerator becomes lesser than denominator.

Again, i want to emphasize that this formuala only works for prime numbers !!
So to find the number of 10's in any n!, DO NOT DIVIDE by 10 ! (10 is not prime !)
i.e DONT do
n/10 + n/100 + n/1000 - THIS IS WRONG !!!
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Re: Math: Number Theory   [#permalink] 25 Sep 2012, 09:43

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