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Number Properties: Tips and hints [#permalink]
04 Jun 2014, 11:25

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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}, ...).

Re: Number Properties: Tips and hints [#permalink]
24 Jun 2014, 22:15

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[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

Re: Number Properties: Tips and hints [#permalink]
25 Jun 2014, 01:42

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Expert's post

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

Please explain.

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).

Number Properties: Tips and hints [#permalink]
27 Aug 2014, 05:37

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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.