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Math: Number Theory

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Re: Math: Number Theory [#permalink] New post 28 Oct 2010, 21:28
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36=6^2=2^2*3^2

Powers of 2 & 3 are even (2)

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Re: Math: Number Theory [#permalink] New post 01 Nov 2010, 15:16
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shrive555 wrote:
If n is a positive integer greater than 1, then there is always a prime number P with n<P<2n

n<p<2n can someone please explain this with example .

Thanks


The result you are referring to is a weak form of what is known as Bertrand's Postulate. The proof of this result is beyond the scope of the GMAT, but it is easy to show some examples.

Choose any n>1, you will always find a prime number between n & 2n.

Eg. n=5, 2n=10 ... p=7 lies in between
n=14, 2n=28 ... p=19 lies in between
n=20, 2n=40 ... p=23 lies in between
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Re: Math: Number Theory [#permalink] New post 27 Oct 2012, 01:34
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About Exponents and divisibility:



(a + b)^2 = a^2+ 2ab + b^2 Square of a Sum
(a - b)^2 = a^2 - 2ab + b^2 Square of a Diffe rence


a^n - b^n is always divisble by a-b i.e. irrespective of n being odd or even
Proof:
a^2 - b^2 = (a-b)(a+b)
a^3 - b^3 = (a-b)(a^2+ab+b^2)

Thus divisible by a- b in both cases where n = 2 i.e. even and 3 i.e. odd

a^n + b^n is divisble by a+b i.e. only if n = odd
Proof:
a^3 - b^3 = (a+b)(a^2-ab+b^2)
Thus divisible by a + b as n = 3 i.e. odd
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Re: Math: Number Theory [#permalink] New post 06 Jan 2010, 14:02
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Bunuel wrote:
---

@Bunuel....

Got hold of an interesting concept of Division of Negative numbers.... as in what would be the quotient and remainder if you we divide -11 by 5.

Please check the post http://gmatclub.com/forum/finding-the-remainder-when-dividing-negative-numbers-88839.html#p670513

May be this concept could be included in the Number Theory post as it seems to be tricky and not that common!

I had learnt about it from this link: http://whyslopes.com/Number_Theory/Decimals_Modular_and_Remainder_Arithmetic.html

Cheers!
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Re: Math: Number Theory [#permalink] New post 24 Jan 2010, 06:33
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no problem. :)

Great post by the way, very informative.
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Re: Math: Number Theory [#permalink] New post 24 Jan 2010, 14:06
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Quote:
ding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.249, since the dropped 5 is equal to 5.


I'm assuming it was just a typo for the last part of the example:
Its entered as 5.3485 rounded to the nearest thousandth = 5.249, since the dropped 5 is equal to 5.


I guess you meant 5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.
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Re: Math: Number Theory [#permalink] New post 25 Jan 2010, 14:54
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I have a question regarding number properties, which I found on an old GMAT test paper form. Here it is:


If the sum of two positive integers is 24 and the difference of their squares is 48, what is the product of the two integers?

(a) 108
(b) 119
(c) 128
(d) 135
(e) 143

Is there a more efficient way of solving this than choosing two numbers at random?
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Re: Math: Number Theory [#permalink] New post 26 Jan 2010, 06:15
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Thank you very much, sorry about that, will do from now on.
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Re: Math: Number Theory [#permalink] New post 26 Jan 2010, 20:01
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Great Post, thanks a lot
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Re: Math: Number Theory [#permalink] New post 05 Mar 2010, 00:22
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Re: Math: Number Theory [#permalink] New post 19 Nov 2010, 00:53
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Araj wrote:
Hello Bunuel - thank you so much for this fantastic post!

with regards to checking for primality:

Quote:
Verifying the primality (checking whether the number is a prime) of a given number can be done by trial division, that is to say dividing by all integer numbers smaller than , thereby checking whether is a multiple of .
Example: Verifying the primality of : is little less than , from integers from to , is divisible by , hence is not prime.


Would it be accurate to say that a number is prime ONLY if it gives a remainder of 1 or 5 when divided by 6?
i.e, for eg. 10973/6 gives a remainder of 5, so it has to be prime...

i found the reasoning behind this in one of the OG solutions:
prime numbers always take the form: 6n+1 or 6n+5 ....

the only possible remainders when any number is divided by 6 are [0,1,2,3,4,5] ...
A prime number always gives a remainder of 1 or 5, because:
a) if the remainder is 2 or 4, then the number must be even
b) if the remainder is 3, then it is divisible by 3 ...

hence, if a number divided by 6 yields 1 or 5 as its remainder, then it must be prime
...?

-Raj



First of all there is no known formula of prime numbers.

Next:
Any prime number p>3 when divided by 6 can only give remainder of 1 or 5 (remainder can not be 2 or 4 as in this case p would be even and remainder can not be 3 as in this case p would be divisible by 3).

So any prime number p>3 could be expressed as p=6n+1 orp=6n+5 or p=6n-1, where n is an integer >1.

But:
Not all number which yield a remainder of 1 or 5 upon division by 6 are prime, so vise-versa of above property is not correct. For example 25 yields a remainder of 1 upon division be 6 and it's not a prime number.

Hope it's clear.
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Re: Math: Number Theory [#permalink] New post 06 Dec 2010, 00:08
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shrive555 wrote:
(a^m)^n=a^{mn} ----------1

(2^2)^2 = 2^2*^2 =2^4

a^m^n=a^{(m^n)} and not (a^m)^n ------------------2

2^2^2 = 2^(2^2) = 2^4

If above example is correct then whats the difference 1 & 2. Please clarify
thanks


If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:
a^m^n=a^{(m^n)} and not (a^m)^n, which on the other hand equals to a^{mn}.

So:
(a^m)^n=a^{mn};

a^m^n=a^{(m^n)} and not (a^m)^n.

Now, there are some specific values of a, m and n for which a^m^n equals to a^{mn}. For example:
a=1: 1^{m^n}=1=1^{mn};

m=0: a^0^n=a^0=1 and a^{0*n}=a^0=1;

m=2 and n=2 --> a^{2^2}=a^4 and a^{2*2}=a^4;

m=4 and n=\frac{1}{2} --> a^{4^{\frac{1}{2}}}=a^2 and a^{4*{\frac{1}{2}}}=a^2;
...

So, generally a^m^n does not equal to (a^m)^n, but for specific values of given variables it does.

shrive555 wrote:
In question would that be given explicitly ... i mean the Brackets ( )


a^m^n ALWAYS means a^{(m^n)}, so no brackets are needed. For example 2^{3^4}=2^{(3^4)}=2^{81};

If GMAT wants the order of operation to be different then the necessary brackets will be put. For example: (2^3)^4=2^{(3*4)}=2^{12}.

Hope it's clear.
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Re: Math: Number Theory [#permalink] New post 03 Jan 2011, 08:52
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resh924 wrote:
Bunuel,

For determining last digit of a power for numbers 0, 1, 5, and 6, I am not clear on how to determine the last digit.

Your post says:
• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.

What is the last digit of 345^27 ---is the last digit 5?
What is the last digit of 216^32----is the last digit 6?
What is the last digit of 111^56---is the last digit 1?

Any clarification would be helpful.

Thanks for all your help.


First of all: last digit of 345^27 is the same as that of 5^27 (the same for 216^32 and 111^56);

Next:
1 in any integer power is 1;
5^1=5, 5^2=25, 5^3=125, ...
6^1=6, 6^2=36, 5^3=216, ...

So yes, integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base: thus 0, 1, 5, and 6 respectively.

Hope it's clear.
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Re: Math: Number Theory [#permalink] New post 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.

Think about this example.
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] New post 26 Jan 2010, 21:57
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Re: Math: Number Theory [#permalink] New post 31 Jan 2010, 09:00
Bunuel wrote:
All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form 6n-1 or 6n+1, because all other numbers are divisible by 2 or 3.


Awesome post, thank you so much! +1

What is the quickest way to figure out whether a number is prime? I usually check if it's odd or even, then sum its digits to figure out if it's divisible by 3, then look if it ends in 5 and if all else fails divide it by 7. Is this the recommended approach?

What might be a bit confusing is that while all prime numbers are of the form 6n-1 or 6n+1, not all numbers of that form are in fact prime. I think this is crucial. For instance, the number 49 is 6n+1, but is not prime.

Any insight on a quicker check (if one exists) would be much appreciated and thank you again for your efforts. They make a real difference!
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Re: Math: Number Theory [#permalink] New post 31 Jan 2010, 09:19
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ariel wrote:
What is the quickest way to figure out whether a number is prime?


Unfortunately, there is no such quick way to say that this number is prime. You can remember all numbers till 50 and then use rule:

Rule: To check whether a number is prime or not, we try to divide it by 2, 3, 5 and so on. You can stop at \sqrt{number} - it is enough. Why? Because if there is prime divisor greater than \sqrt{number}, there must be another prime divisor lesser than \sqrt{number}.

Example,

n = 21 -- > \sqrt{21}~ 4-5
So, we need to check out only 2,3 because for 7, for instance, we have already checked out 3.

n = 101 --> 2,3,5 is out (the last digit is not even or 5 and sum of digits is not divisible by 3). we need to check out only 7
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Re: Math: Number Theory [#permalink] New post 31 Jan 2010, 10:06
Appreciate the very prompt response, walker. To your point re divisibility by 7: I'm having a hard time proving this algebraically, is it a fair statement to say that the only non-prime numbers of the form 6n-1 and 6n+1 are the ones that are divisible by 7?

If so, a quick way to check whether a big number is prime would be to: 1) check whether it's of the form 6n-1 or 6n+1 2) check whether it's divisible by 7

Is this correct?
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Re: Math: Number Theory [#permalink] New post 31 Jan 2010, 10:34
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ariel wrote:
Appreciate the very prompt response, walker. To your point re divisibility by 7: I'm having a hard time proving this algebraically, is it a fair statement to say that the only non-prime numbers of the form 6n-1 and 6n+1 are the ones that are divisible by 7?

If so, a quick way to check whether a big number is prime would be to: 1) check whether it's of the form 6n-1 or 6n+1 2) check whether it's divisible by 7

Is this correct?


Not so. Divisibility by 7 does not check whether the number is prime or not.

Actually this issue is covered in the post. First you should know that all prime numbers except 2 and 5 end in 1, 3, 7 or 9. So if it ends in some other digit it's not prime.

Next, if the above didn't help (meaning that number ends in 1, 3, 7 or 9) there is a way to check whether the number is prime or not. Walker gave an example how to do this, but here it is again:

Verifying the primality of a given number n can be done by trial division, that is to say dividing n by all integer numbers smaller than \sqrt{n}, thereby checking whether n is a multiple of m<\sqrt{n}.

Examples: Verifying the primality of 161: \sqrt{161} is little less than 13. We should check 161 on divisibility by numbers from 2 to 13. From integers from 2 to 13, 161 is divisible by 7, hence 161 is not prime.

Verifying the primality of 149: \sqrt{149} is little more than 12. We should check 149 on divisibility by numbers from 2 to 12, inclusive. 149 is not divisible by any of the integers from 2 to 12, hence 149 is prime.

Verifying the primality of 73: \sqrt{73} is little less than 9. We should check 73 on divisibility by numbers from 2 to 9. 73 is not divisible by any of the integers from 2 to 9, hence 149 is prime.

Hope it helps.
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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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Joined: 06 Oct 2009
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Re: Math: Number Theory [#permalink] New post 01 Feb 2010, 15:46
Hey Bunuel, great post so far, just wondering when the Percent notes will go up in this section.
Re: Math: Number Theory   [#permalink] 01 Feb 2010, 15:46
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1 Experts publish their posts in the topic Number theory..... pzazz12 3 05 Oct 2010, 04:39
33 Experts publish their posts in the topic Math: Number Theory - Percents Bunuel 43 22 Mar 2010, 14:24
8 Experts publish their posts in the topic Math: Number Theory (broken into smaller topics) Bunuel 7 10 Mar 2010, 05:20
NUMBER THEORY vcbabu 5 03 Feb 2009, 10:11
1 NUMBER THEORY vcbabu 2 02 Feb 2009, 10:38
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