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Why is the sum of the reciprocals of all of the divisors of a perfect number equal to 2?
Wiki User
∙ 9y ago
Suppose N is a perfect number. Then N cannot be a square number and so N has an even number of factors.Suppose the factors are f(1) =1, f(2), f(3), ... , f(k-1), f(k)=N.
Furthermore f(r) * f(k+1-r) = N for r = 1, 2, ... k so that f(r) = N/f(k+1-r)
which implies that 1/f(r) = f(k+1-r)/N
Then 1/f(1) + 1/(f(2) + ... + 1/f(k)
= f(k)/N + f(k-1)/N + ... + f(1)/N
= [f(k) + f(k-1) + ... + f(1)] / N
= 2N/N since, by definition, [f(k) + f(k-1) + ... + f(1)] = 2N
Wiki User
∙ 7y ago
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What happens when you divide reciprocals?
A number divided by its reciprocal is equal to the square of the number.
What is the perfect number between 480 and 500?
496 An integer is a perfect number if it is equal to the sum of its positive divisors - not including itself. 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.
How many positive integers less than 2008 have an even number of divisors?
There are 1,963 such integers. Every factor of a number has a pair. The only time there will be an odd number of factors is if one factor is repeated, ie the number is a perfect square. So the question is really asking: how many positive integers less than 2008 (in the range 1 to 2007) are not perfect squares. √2007 = 44 and a bit (it lies between 44 and 45) So there are 44 integers less than (or equal to) 2007 which are perfect squares → 2007 - 44 = 1963 integers are not perfect squares in the range 1-2007 and have an even number of factors (divisors).
Why is the sum of the reciprocals of all of the divisors of a perfect number is equal to 2?
Suppose the perfect number P has divisors f1, f2, ... , fk, P (where all the fs are smaller than P. Then, by the definition of a perfect number, f1 + f2 + ... + fk = P So that f1 + f2 + ... + fk + P = 2P Dividing through by P, f1/P + f2/P + ... + fk/P + P/P = 2 . . . . . . . . . . . (2) Now, since f1 is a factor of P, the f1*g1 = P where g is the cofactor and so f1/P = 1/g1. That is, f1/P is the reciprocal of one of the other factors of P. Also, the last term on the left is the reciprocal of the factor 1. And therefore, f1/P + f2/P + ... + fk/P + P/P = 2 = 1/g1 + 1/g2 + ... + 1/gk + 1/1
Are the reciprocals of a ratio equal?
The reciprocal of a number or a fraction is just flipping it...for example, the recip of 3 is 1/3 or the recip of 4/5 is 5/4
What is a perfect number and give an example of one?
A perfect number is equal to the sum of its proper divisors (the factors excluding the number itself.) 6 is a perfect number. Its proper divisors are 1, 2 and 3.
Why are perfect numbers perfect?
A perfect number is only classified as a perfect number because all of its proper divisors add up to itself. For example, the proper divisors of six are one, two and three. Those numbers added together equal six. Therefore, six is a perfect number.
What are all the almost perfect numbers?
An almost perfect number is a natural number n such that the sum of all divisors of n is equal to 2n - 1.
Why 21 is not a perfect number?
First, you want to know the definition of a perfect number: A perfect number is an integer which is equal to the sum of all its positive integer divisors that are less than itself. Example: The positive integer divisors of 6, less than 6, are 1, 2, and 3. The sum of those divisors is 1 + 2 + 3 = 6. Therefore, 6 is a perfect number. Now, 21 does not meet this definition and therefore is not a perfect number. Let's try it. There are three positive divisors of 21, namely, 1, 3, and 7, that are less than 21, itself. (There are no others.) But the sum of these divisors is 1 + 3 + 7 = 11, which is not equal to 21. Therefore, we confirm that 21 is not a perfect number.
What is the term for equal to the sum of its proper factors?
A perfect number is the term for a number that is equal to the sum of its proper divisors. Be careful not to confuse that with proper factors. Proper divisors include 1 but not the number itself. Proper factors don't include either I or the original number.
What are the perfection numbers?
A number is considered perfect if it is equal to the sum of all its positive factors/divisors, excluding itself. These are the first few perfect numbers: * 6 * 28 * 496 * 8128 * 33550336 * 8589869056A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. An example 1+2+3=6 and 1x2x3=6
What happens when you divide reciprocals?
A number divided by its reciprocal is equal to the square of the number.
Does a reciprocal always equal a whole number?
If you multiply them, reciprocals always equal one.
What are all the perfect numbers 1 - 300?
-299
Why is 496 a perfect number?
Divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and 248 and the sum of all these divisors is 496 (1+2+4+8+16+31+62+124+248=496) so 496 is a perfect number.The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: (1 + 2 + 3 + 6) / 2 = 6.
What is the perfect number between 480 and 500?
496 An integer is a perfect number if it is equal to the sum of its positive divisors - not including itself. 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.
What is a mirror perfect number?
A perfect number is equal to the sum of its proper positive divisors.A mirror perfect number is equal to the sum of its proper positive divisors when they are read backwards.From a paper by Joseph L. Pe which is available on the 'net:10311 = 1 + 3 + 7 + 21 + 491 + 1473 + 34377343 + 3741 + 194 + 12 + 7 + 3 + 1 = 1130110311 is perfect and mirror perfect.
