What are the first 10 perfect numbers?

The first 10 perfect numbers are as follows: 496. 8128. 33550336. 8589869056. 137438691328. 2305843008139952128. More items... •

testbook.com - Perfect Numbers: Definition, List from 1 to 100 & Solved Examples

How to find my passion?

How to find your passion Create a personal vision statement. ... Uncover your values. ... Find your true north. ... Make a list of things you love...

Last updated Dec 29, 2018

What happens when you throw away a hoarders stuff?

Everything has meaning, so by tossing it out you're disrespecting it and the person connected to it." This connection often makes it extremely...

Last updated Mar 10, 2020

Awaken your dormant DNA ability to attract wealth effortlessly

The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.

Learn More »

Promotion

Priyanka Waghmare | Updated: Mar 8, 2022 19:55 IST

We are going to learn the key concepts of what is a perfect number with definition, important formulas, perfect numbers list from 1 to 100 and Euclid’s perfect number theorem. We have also added a few solved examples for the perfect numbers that candidates will find beneficial in their exam preparation. We’ll also learn how to find perfect numbers.

What is a Perfect Number?

A perfect number is a positive integer whose sum of its proper divisors is equal to one. Examples of perfect numbers: Sum of 1, 2, and 3 equals 6, which is the lowest perfect number. The numbers 28, 496, and 8,128 are also perfect. According to number theory, A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.

Perfect Number Definition

The sum of their “proper” divisors equals “perfect numbers” (positive integers that divide a number evenly, not counting itself). Perfect Number Definition 1. The sum of divisors is the function \(σ(n) = \sum_{d|n}d\)᠎ ᠎d, where d runs over the positive divisors of n including 1 and n itself. Example, σ(11) = 1 + 11 = 12 and σ(15) = 1 + 3 + 5 + 15 = 24. Perfect Number Definition 2. The number N is said to be perfect if σ(N) = 2N. When σ(N) < 2N, we say N is deficient; when σ(N) > 2N, we say N is abundant. The definition of perfect number is equivalent to saying that the sum of the proper (or aliquot) divisors of N is equal to N (we just do not add N itself to the sum). While this may seem more natural, the central reason for using the function σ is that it possesses some very special properties.

Euclid’s Perfect Number Theorem

Euclid’s Perfect Number Theorem states that “If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect”.

Perfect Number Example

6 = 3 + 2 + 1, for example, and 28 = 14 + 7 + 4 + 2 + 1. A number is perfect if the sum of its appropriate components equals the number. Make a list of all the numbers that divide a number except the number itself to find the perfect factors. The numbers 1, 2, 3, 6, and 9 are ideal factors for the number 18. If the sum of the elements equals 18, it is a perfect number. 1 + 2 + 3 + 6 + 9 = 6 + 6 + 9 = 12 + 9 = 21

As a result, 18 isn’t perfect.

Perfect Numbers List

The first few perfect numbers \(P_n\) are summarized in the following table together with their corresponding indices p n \(p_n\) \(P_n\) 1 2 6 2 3 28 3 5 496 4 7 8128 5 13 33550336 6 17 8589869056 7 19 137438691328 8 31 2305843008139952128

How do you beat financial depression?

Run Toward Your Financial Problems, Not Away From Them. Create a Financial Plan and Share It With Your Household. Use Positive Language When...

Last updated Jun 10, 2022

What spice whitens your teeth?

turmeric According to some dentists, public figures, and celebrities, turmeric helps whiten teeth. Turmeric is a completely natural product, which...

Last updated Dec 20, 2020

Awaken your dormant DNA ability to attract wealth effortlessly

The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.

Learn More »

Promotion

Perfect Numbers List from 1 to 100

There are only two perfect numbers from 1 to 100. They are 6 and 28.

The first 10 perfect numbers are as follows:

6 28 496 8128 33550336 8589869056 137438691328 2305843008139952128 2658455991569831744654692615953842176 191561942608236107294793378084303638130997321548169216

How to Find Perfect Number?

We can easily find perfect numbers using the following steps:

Perfect numbers are positive integers n such that n=s(n),

where, s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently

sigma(n)=2n,

where sigma(n) is the divisor function (i.e., the sum of divisors of n including n itself). For example, the first few perfect numbers are 6, 28, 496, 8128, … (OEIS A000396), since

6 = 1+2+3

28 = 1+2+4+7+14

496 = 1+2+4+8+16+31+62+124+248,

Perfect Numbers Solved Examples

Solved Example: Find out if 496 is a perfect number.

Solution: The proper factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, and 248 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

496 is perfect.

Hope this article on Perfect Numbers was informative. Get some practice of the same on our free Testbook App. Download Now!

Perfect Numbers FAQs

Q.1 How do you find perfect numbers? Ans.1 Perfect numbers are positive integers n such that n=s(n), where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently sigma(n)=2n, where sigma(n) is the divisor function (i.e., the sum of divisors of n including n itself). For example, the first few perfect numbers are 6, 28, 496, 8128, … (OEIS A000396), since 6 = 1+2+3; 28 = 1+2+4+7+14; 496 = 1+2+4+8+16+31+62+124+248. A number is perfect if the sum of its appropriate components equals the number. Make a list of all the numbers that divide a number except the number itself to find the perfect factors. The numbers 1, 2, 3, 6, and 9 are ideal factors for the number 18. If the sum of the elements equals 18, it is a perfect number Q.2 How to define perfect number? Ans.2 Definition 1. The sum of divisors is the function \(σ(n) = \sum_{d|n}d\) d, where d runs over the positive divisors of n including 1 and n itself. For example, σ(11) = 1 + 11 = 12 and σ(15) = 1 + 3 + 5 + 15 = 24. Definition 2. The number N is said to be perfect if σ(N) = 2N. When σ(N) < 2N, we say N is deficient; when σ(N) > 2N, we say N is abundant. The definition of perfect is equivalent to saying that the sum of the proper or aliquot) divisors of N is equal to N (we just do not add N itself to the sum). While this may seem more natural, the central reason for using the function σ is that possesses some very special properties. Q.3 What is a perfect number example? Ans.3 6 = 3 + 2 + 1, for example, and 28 = 14 + 7 + 4 + 2 + 1. A number is perfect if the sum of its appropriate components equals the number. Make a list of all the numbers that divide a number except the number itself to find the perfect factors. The numbers 1, 2, 3, 6, and 9 are ideal factors for the number 18. If the sum of the elements equals 18, it is a perfect number. 1 + 2 + 3 + 6 + 9 = 6 + 6 + 9 = 12 + 9 = 21 As a result, 18 isn’t perfect. Q.4 Why is 10 a perfect number? Ans.4 A perfect number is usually defined in terms of its divisors, while early definitions were defined in terms of a number’s ‘aliquot components.’ A suitable quotient of a number is an aliquot fraction of the number. The aliquot parts of ten, for example, are 1, 2, and 5…. The result is 496, which is a perfect number. Q.5 Why 6 is a perfect number? Ans.5 A perfect number is a positive integer whose sum of its proper divisors is equal to one. The sum of 1, 2, and 3 equals 6, which is the lowest perfect number. 6 = 3 + 2 + 1, for example, and 28 = 14 + 7 + 4 + 2 + 1. A number is perfect if the sum of its appropriate components equals the number. Make a list of all the numbers that divide a number except the number itself to find the perfect factors. The numbers 1, 2, 3, 6, and 9 are ideal factors for the number 18. If the sum of the elements equals 18, it is a perfect number. 1 + 2 + 3 + 6 + 9 = 6 + 6 + 9 = 12 + 9 = 21 As a result, 18 isn’t perfect. Q.6 What is a Perfect Number? Ans.6 A perfect number is a positive integer whose sum of its proper divisors is equal to one. The sum of 1, 2, and 3 equals 6, which is the lowest perfect number. The numbers 28, 496, and 8,128 are also perfect. A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself, according to number theory. Pythagoras and his followers studied perfect numbers more for their mystical characteristics than for their mystical properties than for their number theoretic properties.

Get Daily GK & Current Affairs Capsule & PDFs

What qualifies as house poor?

The expressions “house poor” and “house broke” refer to the situation where homeowners have bought homes beyond their means. They end up spending...

Last updated Feb 7, 2019

What are the 8 types of magic?

There are eight classic schools of magic in Dungeons & Dragons, as originally named in Advanced Dungeons & Dragons: abjuration, alteration,...

Last updated Nov 5, 2022

Awaken your dormant DNA ability to attract wealth effortlessly

The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.

Learn More »

Promotion

Get 100+ Free Mock Tests & Quizzes

testbook.com - Perfect Numbers: Definition, List from 1 to 100 & Solved Examples

What are the Native American zodiac signs?

Red Hawk And Aries. People born at this time have a strong part of them that is passionate and full of energy. ... Beaver And Taurus. ... Deer And...

Last updated Jul 29, 2018

What is the lucky number for 2022?

6 and 8 are the two luckiest numbers for 2022. 6 pronounces similar to the Chinese expressions of “smooth”, so 66 means “everything goes well”. 8...

Last updated Jan 21, 2018

Awaken your dormant DNA ability to attract wealth effortlessly

The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.

Learn More »

Promotion

Can too much cinnamon damage liver?

Cinnamon Risks While this spice has many favourable aspects, there is a downside to most types of cinnamon: coumarin. Coumarin is a chemical...

Last updated Feb 5, 2018

Awaken your dormant DNA ability to attract wealth effortlessly

The simple yet scientifically proven Wealth DNA method laid out in the report allows you to effortlessly start attracting the wealth and abundance you deserve.

Learn More »

Promotion

What can worsen cholesterol?

Factors that can increase your risk of unhealthy cholesterol levels include: Poor diet. Eating too much saturated fat or trans fats can result in...

Last updated Feb 26, 2022