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Math: Number Theory

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Math Expert
Joined: 02 Sep 2009
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16 Aug 2015, 10:11
zmtalha wrote:
In Roots section you wrote:

$$\sqrt{x^2}$$=|x|, when x0, then, $$\sqrt{x^2}$$=-x and when x≥0, then $$\sqrt{x^2}$$=x.

Here, won't it be "when x<0, then $$\sqrt{x^2}$$=−x"?

No. For x = 0, we'd get $$\sqrt{0^2}=0=-0$$.
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18 Aug 2015, 14:03
SQUINGEL wrote:
how do you get the 13 and the 11 in the question?
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?

x+y=24
and
x2−y2=48 --> (x+y)(x−y)=48, as x+y=24 --> 24(x−y)=48 --> x−y=2 --> solving for x and y --> x=13 and y=11 --> xy=143.

We have two equations:
x + y = 24;
x - y = 2.

Sum those two:
(x + y) + (x - y) = 24 + 2
2x = 26
x = 13

Substitute x = 13 into any of the equations:
13 - y = 2
y = 11.
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23 Aug 2015, 21:40
I have a theory-related question regarding the definition of intersecting lines. If a line overlaps (colinear) a line segment (more than one point, of course) are they still considered to be "intersecting"? I was under the impression that this would not constitute an intersection however a question I worked seemed to suggest the opposite.

Example: line segment (1,5),(3,3) and line y=-x+6

The line overlaps the line segment; are they "intersecting"?
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24 Aug 2015, 08:42
ar500 wrote:
I have a theory-related question regarding the definition of intersecting lines. If a line overlaps (colinear) a line segment (more than one point, of course) are they still considered to be "intersecting"? I was under the impression that this would not constitute an intersection however a question I worked seemed to suggest the opposite.

Example: line segment (1,5),(3,3) and line y=-x+6

The line overlaps the line segment; are they "intersecting"?

Technically intersect means share one or more points in common. So, if two lines overlap they do intersect.

Having said that, I must add that GMAT would never test you on such technicalities, you can ignore this question and move on.

Hope it helps.
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14 Sep 2016, 07:02
ks4196 wrote:
Quote:
Similarly, all prime numbers above 3 are of the form 6n−16n−1 or 6n+16n+1, because all other numbers are divisible by 2 or 3.
In the prime number section it says prime numbers greater than 3 are of the form 6n-1 or 6n+1. However, this is not necessarily true. eg: n=36, then 6n-1 is 215 and 6n+1 is 217, divisible by 5 and 7 respectively.

That's not what is said there.

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.
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27 Oct 2016, 02:30
• Any positive divisor of n is a product of prime divisors of n raised to some power.

pls someone explain with example
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27 Oct 2016, 03:59
sanaexam wrote:
• Any positive divisor of n is a product of prime divisors of n raised to some power.

pls someone explain with example

For example, say n = 72. Consider it's factor 36 --> $$36 = 2^2*3^2$$ --> 36 = product of prime divisors of n raised to some power.
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23 May 2017, 16:47
Bunuel wrote:
NUMBER THEORY

Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....).

--------------------------------------------------------

0.333333... looks like a recurring decimal and is not a rational number. This can be better modified as 0.33333 etc so it is less confusing in my opinion.
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24 May 2017, 04:02
workout wrote:
Bunuel wrote:
NUMBER THEORY

Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....).

--------------------------------------------------------

0.333333... looks like a recurring decimal and is not a rational number. This can be better modified as 0.33333 etc so it is less confusing in my opinion.

0.333333... IS a rational number because it equals to the ratio of two integers: 1/3 = 0.3333......
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27 Mar 2018, 05:40
Hi Bunuel
The topic doesn’t touch up on LCM and HCF for fractions..

LCM( (a/b) , (c/d) ) = LCM(a,c)/HCF(b,d))

HCF( (a/b) , (c/d) ) = HCF(a,c)/LCM(b,d)

Question from where I started to look for the theory:
https://gmatclub.com/forum/what-is-the- ... ml#p818818
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04 May 2018, 04:45
Bunuel niks18 gmatbusters pushpitkc VeritasPrepKarishma

I have a small query regarding rounding:

How do I interpret nearest ten, nearest hundred etc in

1234.1234

on both LHS and RHS of decimal?
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04 May 2018, 06:56
gmatbusters

Thanks for the pic , that made me wonder if the difference in position of tens on LHS and RHS
is because of raising base of 10 to exponents say 0,1 and -1. Is this how we arrive at
PLACE VALUES for a number?

Sorry, coming from engineering bakground, am still
not able to erase why aspects of logic

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06 Oct 2018, 22:02
I just started looking at this and I'm impressed. I have a suggestion for the LCM section. I believe it should be changed as follows, but please correct me if I'm wrong...
"Lowest Common Multiple - LCM

The lowest common multiple orlowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.

To find the LCM, you will need to do prime-factorization. Then multiply all the factors. (For any factors that are common, use the highest power.)"
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01 Dec 2018, 00:13
Can someone please explain me the last digit concept using more examples?

LAST DIGIT OF A PRODUCT
Last digits of a product of integers are last digits of the product of last digits of these integers.
For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=540*104=40*4=160=60
Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?
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09 Dec 2018, 09:55
Hi. Is it possible to solve official questions related to this specific topic "Numbers Theory", after going over this topic here?

So, for instance, let's say we read the topic "Percents" from the below link. Ideally, want to solve questions related to each topic as I study through. Just want to know if something like that is a possibility on the forum.

https://gmatclub.com/forum/all-you-need ... l#p1130136

Cheers.
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08 Jan 2019, 07:15
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.

Hi

Can some expert please explain this concept more clearly.
What I am looking for is the proof of these statements.

Bunuel
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29 Mar 2019, 01:11
• If a is a factor of b and b is a factor of a, then a=b or a=−b.

I did not understand this. Could you please explain?
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29 Mar 2019, 01:44
It means a/b is an integer
Also, b/a is an integer.

This is only possible if either a=b or a=-b.

Hope, it is clear now.

dee1711s wrote:
• If a is a factor of b and b is a factor of a, then a=b or a=−b.

I did not understand this. Could you please explain?

Posted from my mobile device
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02 Apr 2019, 00:28
• If a number equals the sum of its proper divisors, it is said to be a perfect number.
Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:
• If aa is a factor of bb and aa is a factor of cc, then aa is a factor of (b+c)(b+c). In fact, aa is a factor of (mb+nc)(mb+nc) for all integers mm and nn.

• If aa is a factor of bb and bb is a factor of cc, then aa is a factor of cc.

• If aa is a factor of bb and bb is a factor of aa, then a=ba=b or a=−ba=−b.

• If aa is a factor of bcbc, and gcd(a,b)=1gcd(a,b)=1, then a is a factor of cc.

• If pp is a prime number and pp is a factor of abab then pp is a factor of aa or pp is a factor of bb.

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02 Apr 2019, 00:29
• If a number equals the sum of its proper divisors, it is said to be a perfect number.
Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:
• If aa is a factor of bb and aa is a factor of cc, then aa is a factor of (b+c)(b+c). In fact, aa is a factor of (mb+nc)(mb+nc) for all integers mm and nn.

• If aa is a factor of bb and bb is a factor of cc, then aa is a factor of cc.

• If aa is a factor of bb and bb is a factor of aa, then a=ba=b or a=−ba=−b.

• If aa is a factor of bcbc, and gcd(a,b)=1gcd(a,b)=1, then a is a factor of cc.

• If pp is a prime number and pp is a factor of abab then pp is a factor of aa or pp is a factor of bb.

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