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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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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0.333333... IS a rational number because it equals to the ratio of two integers: 1/3 = 0.3333......
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Adding Few Formulas

Sum of First "n" natural nos = n(n+1)/2
Sum of First "n" ODD natural nos = n^2
Sum of First "n" EVEN natural nos = n (n+1)
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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)

Please add if tested on GMAT.

Question from where I started to look for the theory:
https://gmatclub.com/forum/what-is-the- ... ml#p818818

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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Hii

You have asked regarding rounding to nearest tens, hundred for 1234.1234 on both LHS and RHS of decimal.

First of all , Lets know what does the place signify:
we denote the digits as units, tens, hundreds, thousands before decimal and tenth, hundredth after decimal. see the sketch

Rounding
Simplifying a number to a certain place value. 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.

It means if we round 47 to nearest ten = 50
whereas, 44 to nearest ten = 40

Now lets come to your question:
1234.1234
Rounding to nearest ten = 1230
Rounding to nearest hundred = 1200
Rounding to nearest tenth =1234.1
Rounding to nearest hundredth= 1234.12

I hope it is clear now. Fell free to tag again.



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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?


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yes you got it right


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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.)"

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?

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

Thanks in advance.

Cheers.

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
VeritasKarishma
chetan2u
Gladiator59
gmatbusters
MathRevolution
AjiteshArun

Hi nitesh,

It is to do with binomial theorem, which further deals with expansion of a term..
Say you are looking for a^n . I can write a = a-b+b..
\(a^n=(a-b+b)^n=((a-b)+b)^n = (a-b)^n+n(a-b)^{n-1}b^1+....+b^n.....a^n-b^n=(a-b)^n+n(a-b)^{n-1}b^1+...=(a-b)((a-b)^{n-1}+.........)\)
So, Right hand side is multiple of a-b and on left side we have a^n-b^n..
so a^n-b^n is a multiple of a-b

similarly for the other too..

Just take small values to confirm..

Let n = 4.. \(a^4-b^4=(a^2-b^2)(a^2+b^2)=(a-b)(a+b)(a^2+b^2)\).. so multiple of a-b and a+b.
Let n = 3... \(a^3-b^3=(a-b)(a^2+ab+b^2)\)... so multiple of only a-b

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• 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?

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.



Posted from my mobile device
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• 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.


PLEASE CAN YOU EXPLAIN ABOUT THE ABOVE WITH EXAMPLES?

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Explained on previous pages.
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