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Math Expert V
Joined: 02 Sep 2009
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Number Properties: Tips and hints

 ! This post is a part of the Quant Tips and Hints by Topic Directory focusing on Quant topics and providing examples of how to approach them. Most of the questions are above average difficulty.

EVEN/ODD
1. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder.
2. An odd number is an integer that is not evenly divisible by 2.
3. According to the above both negative and positive integers can be even or odd.

ZERO:
1. 0 is an integer.
2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.
3. 0 is neither positive nor negative integer (the only one of this kind).
4. 0 is divisible by EVERY integer except 0 itself, (or, which is the same, zero is a multiple of every integer except zero itself).

PRIME NUMBERS:
1. 1 is not a prime, since it only has one divisor, namely 1.
2. Only positive numbers can be primes.
3. There are infinitely many prime numbers.
4. the only even prime number is 2. Also 2 is the smallest prime.
5. All prime numbers except 2 and 5 end in 1, 3, 7 or 9.

PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);
4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

IRRATIONAL NUMBERS
1. An irrational number is any real number that cannot be expressed as a ratio of integers.
2. The square root of any positive integer is either an integer or an irrational number. So, $$\sqrt{x}=\sqrt{integer}$$ cannot be a fraction, for example it cannot equal to 1/2, 3/7, 19/2, ... It MUST be an integer (0, 1, 2, 3, ...) or irrational number (for example $$\sqrt{2}$$, $$\sqrt{3}$$, $$\sqrt{17}$$, ...).

This week's PS question
This week's DS Question

Theory on Number Properties: http://gmatclub.com/forum/math-number-theory-88376.html

Number Properties GMAT Questions - Master List:
DS Number Properties Problems to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=38
PS Number Properties Problems to practice: http://gmatclub.com/forum/search.php?se ... &tag_id=59

Please share your number properties tips below and get kudos point. Thank you.
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
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Tanvr wrote:
[wrapimg=]PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);[/wrapimg]

I was learning the above points, while a confusion rose. I'd be grateful if you could explain.

We know that 100 is a perfect square (10^2)
If we Prime factorize 100, we get

100= 2*2*5*5

number 1 tips says that a perfect square has odd number of distinct factors. Here 100 has two distinct factors (2,5). How that can be explained ?

number 3 tips says that a perfect square always has an odd number of odd factors and even number of even factors. Here we see, 100 has two 5's which is even number.

Even if we see number 3 tips as, a perfect square always has an odd number of DISTINCT odd factors and even number of DISTINCT even factors., we find that, 100 has only ONE DISTINCT even factor which is 2

2 and 5 are prime factors of 100. The total number of factors of 100=2^2*5^2 is (2+1)(2+1)=9=odd: 1, 2, 4, 5, 10, 20, 25, 50, 100. Out of these 9 factors three are odd (1, 5, and 25) and 6 are even (2, 4, 10, 20, 50, 100).

Hope it helps.
_________________
Intern  Joined: 27 Aug 2014
Posts: 27
GMAT Date: 09-27-2014
Re: Number Properties: Tips and hints  [#permalink]

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15
Another tip,

If B is 1/x times more than A, then A is 1/(x+1) times lesser than B.

This is especially useful in averages, profit and loss, time rate questions.

Example:

If B's wage is 25% more than A's wage, then what is A's wage in terms of B?

B is 1/4 times more than A, so A will be 1/5 or 20% lesser than B. i.e., A = 80% of B
##### General Discussion
Intern  Status: Into the last good fight
Joined: 22 Feb 2012
Posts: 15
GMAT 1: 550 Q42 V24
GMAT 2: 550 Q44 V23
GMAT 3: 540 Q41 V24
Re: Number Properties: Tips and hints  [#permalink]

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3
[wrapimg=]PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);[/wrapimg]

I was learning the above points, while a confusion rose. I'd be grateful if you could explain.

We know that 100 is a perfect square (10^2)
If we Prime factorize 100, we get

100= 2*2*5*5

number 1 tips says that a perfect square has odd number of distinct factors. Here 100 has two distinct factors (2,5). How that can be explained ?

number 3 tips says that a perfect square always has an odd number of odd factors and even number of even factors. Here we see, 100 has two 5's which is even number.

Even if we see number 3 tips as, a perfect square always has an odd number of DISTINCT odd factors and even number of DISTINCT even factors., we find that, 100 has only ONE DISTINCT even factor which is 2

Intern  Joined: 07 Jul 2014
Posts: 16
Re: Number Properties: Tips and hints  [#permalink]

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1
1
Can I suggest a few new topics? Specifically Geometry, word problems, and combinatrics/probability?
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: Number Properties: Tips and hints  [#permalink]

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tangt16 wrote:
Can I suggest a few new topics? Specifically Geometry, word problems, and combinatrics/probability?

Thank you. Will cover them soon.
_________________
Intern  Joined: 27 Aug 2014
Posts: 27
GMAT Date: 09-27-2014
Re: Number Properties: Tips and hints  [#permalink]

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3
If progressions comes under this topic, would like to add this tip.

In the specific case of sum to n1 terms being equal to sum to n2 terms of the same arithmetic progression, the sum of the term numbers which exhibit equal sums is constant for the given evenly spaced set of numbers.

(S3 denotes Sum of the first three terms of the evenly spaced set.)

1.
Q: if sum to 11 terms equal sum to 19 terms in an evenly spaced set, what is the sum to 30 terms for this series?
A: S11 = S19; so S0 = S30. Since S0 = 0, S30 = 0.

2.
This happens because the arithmetic progression's negative terms cancel out the positive terms.

Also, if the series has a zero in it, the sum will be equal for two terms such that one term number will be odd and the other will be even.
Ex.: -10, -5, 0, 5, 10.....

And if the series does not have a zero in it, the sum will be equal for two terms such that both term numbers will be either odd or even.

Ex.: -12, -4, 4, 12......
Intern  Joined: 27 Aug 2014
Posts: 27
GMAT Date: 09-27-2014
Re: Number Properties: Tips and hints  [#permalink]

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5
Tip for questions involving recurring decimals:

Note the following pattern for repeating decimals:
0.22222222... = 2/9
0.54545454... = 54/99
0.298298298... = 298/999

Note the pattern if zeroes preceed the repeating decimal:
0.022222222... = 2/90
0.00054545454... = 54/99000
0.00298298298... = 298/99900
Director  G
Joined: 02 Sep 2016
Posts: 621
Re: Number Properties: Tips and hints  [#permalink]

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Where can I access the file for geometry and word problems?
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: Number Properties: Tips and hints  [#permalink]

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1
Shiv2016 wrote:
Where can I access the file for geometry and word problems?

Go through the following topic ALL YOU NEED FOR QUANT ! ! !.

Hope it helps.
_________________
Director  G
Joined: 02 Sep 2016
Posts: 621
Re: Number Properties: Tips and hints  [#permalink]

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1
Bunuel wrote:
Shiv2016 wrote:
Where can I access the file for geometry and word problems?

Go through the following topic ALL YOU NEED FOR QUANT ! ! !.

Hope it helps.

Great compilation. Thanks Bunuel.
Intern  B
Joined: 29 Aug 2016
Posts: 20
GMAT 1: 490 Q36 V22 GPA: 3
Re: Number Properties: Tips and hints  [#permalink]

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Manager  B
Joined: 20 Feb 2017
Posts: 66
Re: Number Properties: Tips and hints  [#permalink]

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Hi Bunuel,

I have a query regarding the prime numbers. In DS question if it is told that numbers are consecutive prime numbers then from where we have to start numbers?

Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Number Properties: Tips and hints  [#permalink]

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Manager  B
Joined: 01 Apr 2020
Posts: 78
Location: India
Re: Number Properties: Tips and hints  [#permalink]

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Bakervai wrote:
[wrapimg=]PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);[/wrapimg]

I was learning the above points, while a confusion rose. I'd be grateful if you could explain.

We know that 100 is a perfect square (10^2)
If we Prime factorize 100, we get

100= 2*2*5*5

number 1 tips says that a perfect square has odd number of distinct factors. Here 100 has two distinct factors (2,5). How that can be explained ?

number 3 tips says that a perfect square always has an odd number of odd factors and even number of even factors. Here we see, 100 has two 5's which is even number.

Even if we see number 3 tips as, a perfect square always has an odd number of DISTINCT odd factors and even number of DISTINCT even factors., we find that, 100 has only ONE DISTINCT even factor which is 2

Were you able to figure it out? I have the same question!!
Manager  B
Joined: 01 Apr 2020
Posts: 78
Location: India
Re: Number Properties: Tips and hints  [#permalink]

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Bunuel wrote:

Number Properties: Tips and hints

PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

According to (1) and (2)
36 has ODD number of distinct factors...
but discrinct factors of 36 are 2 and 3
i.e. TWO factors i.e. EVEN.

Also, if you are going to include 1 in the list, it makes statement (2) Flawed!!
Because
(A) 1 + 2 + 3 = 6 = Even ≠ Odd

Shouldn't statement (1) use the term Distinct Prime factors?

I'm confused!
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: Number Properties: Tips and hints  [#permalink]

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1
D4kshGargas wrote:
Bakervai wrote:
[wrapimg=]PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);[/wrapimg]

I was learning the above points, while a confusion rose. I'd be grateful if you could explain.

We know that 100 is a perfect square (10^2)
If we Prime factorize 100, we get

100= 2*2*5*5

number 1 tips says that a perfect square has odd number of distinct factors. Here 100 has two distinct factors (2,5). How that can be explained ?

number 3 tips says that a perfect square always has an odd number of odd factors and even number of even factors. Here we see, 100 has two 5's which is even number.

Even if we see number 3 tips as, a perfect square always has an odd number of DISTINCT odd factors and even number of DISTINCT even factors., we find that, 100 has only ONE DISTINCT even factor which is 2

Were you able to figure it out? I have the same question!!

1. Divisors of 100 are: 1 | 2 | 4 | 5 | 10 | 20 | 25 | 50 | 100 (9 divisors).

2. 100 has 3 odd factors (1, 5, and 25) and 6 even factors (2, 4, 10, 20, 50 and 100)
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: Number Properties: Tips and hints  [#permalink]

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D4kshGargas wrote:
Bunuel wrote:

Number Properties: Tips and hints

PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

According to (1) and (2)
36 has ODD number of distinct factors...
but discrinct factors of 36 are 2 and 3
i.e. TWO factors i.e. EVEN.

Also, if you are going to include 1 in the list, it makes statement (2) Flawed!!
Because
(A) 1 + 2 + 3 = 6 = Even ≠ Odd

Shouldn't statement (1) use the term Distinct Prime factors?

I'm confused!

Factors of 36 are 1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 36 (9 divisors).
Odd factors of 36 are 1, 3, and 9.
Even factors of 36 are 2, 4, 6, 12, 18, and 26.
Prime factors of 36 are 2 and 3: 36 = 2^2*3^2.
_________________
Manager  B
Joined: 01 Apr 2020
Posts: 78
Location: India
Re: Number Properties: Tips and hints  [#permalink]

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Bunuel wrote:
D4kshGargas wrote:
Bunuel wrote:

Number Properties: Tips and hints

PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

According to (1) and (2)
36 has ODD number of distinct factors...
but discrinct factors of 36 are 2 and 3
i.e. TWO factors i.e. EVEN.

Also, if you are going to include 1 in the list, it makes statement (2) Flawed!!
Because
(A) 1 + 2 + 3 = 6 = Even ≠ Odd

Shouldn't statement (1) use the term Distinct Prime factors?

I'm confused!

Factors of 36 are 1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 36 (9 divisors).
Odd factors of 36 are 1, 3, and 9.
Even factors of 36 are 2, 4, 6, 12, 18, and 26.
Prime factors of 36 are 2 and 3: 36 = 2^2*3^2.

Hi, thanks for the response.
Just to clear things out,
The term “Distinct Factors” means all the Integers that are part of the Set of factors, whether prime or not.
Right?

I think I was unnecessarily confusing myself between distinct factors and distinct prime factors.
Thanks Math Expert V
Joined: 02 Sep 2009
Posts: 64891
Re: Number Properties: Tips and hints  [#permalink]

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D4kshGargas wrote:
Bunuel wrote:
D4kshGargas wrote:
Bunuel wrote:

Number Properties: Tips and hints

PERFECT SQUARES
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

According to (1) and (2)
36 has ODD number of distinct factors...
but discrinct factors of 36 are 2 and 3
i.e. TWO factors i.e. EVEN.

Also, if you are going to include 1 in the list, it makes statement (2) Flawed!!
Because
(A) 1 + 2 + 3 = 6 = Even ≠ Odd

Shouldn't statement (1) use the term Distinct Prime factors?

I'm confused!

Factors of 36 are 1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 36 (9 divisors).
Odd factors of 36 are 1, 3, and 9.
Even factors of 36 are 2, 4, 6, 12, 18, and 26.
Prime factors of 36 are 2 and 3: 36 = 2^2*3^2.

Hi, thanks for the response.
Just to clear things out,
The term “Distinct Factors” means all the Integers that are part of the Set of factors, whether prime or not.
Right?

I think I was unnecessarily confusing myself between distinct factors and distinct prime factors.
Thanks Yes, prime factors and factors are not the same. For example, factors of 6 are 1, 2, 3, and 6. While prime factors of 6 are 2 and 3.
_________________ Re: Number Properties: Tips and hints   [#permalink] 01 May 2020, 09:31

# Number Properties: Tips and hints  