It is currently 20 Nov 2017, 20:08

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Math: Number Theory

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132851 [0], given: 12378

### Show Tags

14 Sep 2010, 06:38
cheetarah1980 wrote:
There seems to be a discrepancy in what some study guides consider consecutive integers. In Kaplan Premier 2011 consecutive integers are defined as numbers that occur at a fixed interval or exhibit a fixed pattern. However, on the Kaplan Free Practice Test I got a DS question wrong because it didn't consider evenly spaced numbers to necessarily be consecutive. Your definition also separates the two. Could anyone clarify which is correct so I know for the actual GMAT. Thanks!

When we see "consecutive integers" it ALWAYS means integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ...

-7, -6, -5 are consecutive integers.

2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.

3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.

2, 5, 8, 11 ARE NOT consecutive integers, they are terms of arithmetic progression with common difference of 3.

All sets of consecutive integers represent arithmetic progression but not vise-versa.

Hope it's clear.
_________________

Kudos [?]: 132851 [0], given: 12378

 Kaplan GMAT Prep Discount Codes Math Revolution Discount Codes Veritas Prep GMAT Discount Codes
Intern
Joined: 27 Aug 2010
Posts: 23

Kudos [?]: 20 [0], given: 2

### Show Tags

20 Sep 2010, 00:27
7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

Is this the only way to check divisibility by 7? For huge numbers there is no big difference to divide the number directly by 7 or to use the algorithm above.

Kudos [?]: 20 [0], given: 2

Math Expert
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132851 [0], given: 12378

### Show Tags

20 Sep 2010, 00:35
Kronax wrote:
7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

Is this the only way to check divisibility by 7? For huge numbers there is no big difference to divide the number directly by 7 or to use the algorithm above.

Note that you can perform this operation number of times. Also you won't need to check divisibility by 7 for huge numbers on GMAT.
_________________

Kudos [?]: 132851 [0], given: 12378

Intern
Joined: 27 Aug 2010
Posts: 23

Kudos [?]: 20 [0], given: 2

### Show Tags

20 Sep 2010, 00:45
Bunuel wrote:
Kronax wrote:
7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

Is this the only way to check divisibility by 7? For huge numbers there is no big difference to divide the number directly by 7 or to use the algorithm above.

Note that you can perform this operation number of times. Also you won't need to check divisibility by 7 for huge numbers on GMAT.

Thank you for the notice! I haven't thought about doing the operation multiple times.

Kudos [?]: 20 [0], given: 2

Manager
Joined: 20 Apr 2010
Posts: 209

Kudos [?]: 90 [0], given: 28

Schools: ISB, HEC, Said

### Show Tags

29 Sep 2010, 23:30
How to print this document

Kudos [?]: 90 [0], given: 28

Senior Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 312

Kudos [?]: 602 [0], given: 193

### Show Tags

28 Oct 2010, 18:42
Quote:
Perfect Square

A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square.

There are some tips about the perfect square:.
Perfect square always has even number of powers of prime factors.

Bunuel : can you please give an example of bold statement

Thanks
_________________

I'm the Dumbest of All !!

Kudos [?]: 602 [0], given: 193

Retired Moderator
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1209 [2], given: 25

Location: London

### Show Tags

28 Oct 2010, 22:28
2
KUDOS
$$36=6^2=2^2*3^2$$

Powers of 2 & 3 are even (2)

Posted from my mobile device
_________________

Kudos [?]: 1209 [2], given: 25

Senior Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 312

Kudos [?]: 602 [0], given: 193

### Show Tags

01 Nov 2010, 11:09
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
_________________

I'm the Dumbest of All !!

Kudos [?]: 602 [0], given: 193

Retired Moderator
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1209 [2], given: 25

Location: London

### Show Tags

01 Nov 2010, 16:16
2
KUDOS
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
_________________

Kudos [?]: 1209 [2], given: 25

Senior Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 312

Kudos [?]: 602 [0], given: 193

### Show Tags

02 Nov 2010, 08:51
_________________

I'm the Dumbest of All !!

Kudos [?]: 602 [0], given: 193

Intern
Joined: 17 Nov 2009
Posts: 1

Kudos [?]: [0], given: 0

### Show Tags

06 Nov 2010, 13:36
Bunnel!!! you are the rock !
Many thanks for the above post !! It helps me a lot since my exam is very soon !!

Kudos [?]: [0], given: 0

Manager
Joined: 01 Nov 2010
Posts: 173

Kudos [?]: 52 [0], given: 20

Location: Zürich, Switzerland

### Show Tags

14 Nov 2010, 13:32
Thanks once again Bunuel...

Kudos [?]: 52 [0], given: 20

Intern
Joined: 18 Oct 2010
Posts: 3

Kudos [?]: 8 [0], given: 1

### Show Tags

18 Nov 2010, 23:47
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

Kudos [?]: 8 [0], given: 1

Math Expert
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132851 [1], given: 12378

### Show Tags

19 Nov 2010, 01:53
1
KUDOS
Expert's post
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 $$p=6n+1$$ or$$p=6n+5$$ 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.
_________________

Kudos [?]: 132851 [1], given: 12378

Intern
Joined: 18 Oct 2010
Posts: 3

Kudos [?]: 8 [0], given: 1

### Show Tags

19 Nov 2010, 02:18
Quote:
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.

Understood Sir!
.. i'll just use it one way; i.e, if i'm told that n is a prime number>3, then i can express it as 6n+1 or 6n+5

I think I just got a bit too excited about it that I forgot to thoroughly test it thru...

thx again for the prompt reply!

Raj

Kudos [?]: 8 [0], given: 1

Senior Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 312

Kudos [?]: 602 [0], given: 193

### Show Tags

05 Dec 2010, 16:36
$$(a^m)^n=a^{mn}$$ ----------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
_________________

I'm the Dumbest of All !!

Kudos [?]: 602 [0], given: 193

Retired Moderator
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1209 [0], given: 25

Location: London

### Show Tags

05 Dec 2010, 20:36
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

I think its just that you have taken a bad example here. Consider a=2, m=3, b=2

$$(a^m)^n=(2^3)^2=8^2=64$$
$$a^{(m^n)}=2^{(3^2)}=2^9=512$$
_________________

Kudos [?]: 1209 [0], given: 25

Senior Manager
Status: Do and Die!!
Joined: 15 Sep 2010
Posts: 312

Kudos [?]: 602 [0], given: 193

### Show Tags

05 Dec 2010, 22:16
shrouded1 wrote:
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

I think its just that you have taken a bad example here. Consider a=2, m=3, b=2

$$(a^m)^n=(2^3)^2=8^2=64$$
$$a^{(m^n)}=2^{(3^2)}=2^9=512$$

In question would that be given explicitly ... i mean the Brackets ( )
_________________

I'm the Dumbest of All !!

Kudos [?]: 602 [0], given: 193

Math Expert
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132851 [1], given: 12378

### Show Tags

06 Dec 2010, 01:08
1
KUDOS
Expert's post
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.
_________________

Kudos [?]: 132851 [1], given: 12378

Manager
Joined: 21 Jul 2010
Posts: 55

Kudos [?]: 30 [0], given: 3

### Show Tags

30 Dec 2010, 10:39
Thanks for this. It made my life easier! Kudossssssss!

Kudos [?]: 30 [0], given: 3

Re: Math: Number Theory   [#permalink] 30 Dec 2010, 10:39

Go to page   Previous    1   2   3   4   5   6   7   8   9   10   11    Next  [ 202 posts ]

Display posts from previous: Sort by