• Any positive divisor of n is a product of prime divisors of n raised to some power.



pls someone explain with example

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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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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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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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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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.)"

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.

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?

• 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?