À propos de Calculateur de diviseurs : tous les diviseurs d’un nombre
A factor of n is a whole number that divides n with no remainder, and factors come in pairs d × (n ÷ d). The calculator finds the prime factorization first, by trial division and Pollard's rho method, and builds every factor from it: a factor of 48 = 2⁴ × 3 uses 0 to 4 twos and 0 or 1 three, which gives (4 + 1)(1 + 1) = 10 factors.
The same factorization gives the sum of the factors. Subtracting n leaves the sum of the proper factors, which decides the class that goes back to Euclid and Nicomachus: perfect when it equals n (28 = 1 + 2 + 4 + 7 + 14), abundant when it is larger (48 has 76) and deficient when it is smaller, as every prime is.
Numbers up to 10¹⁸ are accepted and factored in a fraction of a second. Up to 5,000 factors are listed in full; beyond that only their count and sum are given. For a negative number the factors come in ± pairs.
Questions
How do you find all the factors of a number?
Test each whole number from 1 up to the square root; every divisor d you find brings its partner n ÷ d. For 48 the square root is about 6.9, and 1, 2, 3, 4 and 6 divide it, giving the pairs 1 × 48, 2 × 24, 3 × 16, 4 × 12 and 6 × 8. That is 10 factors: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. For large numbers, factor into primes first and combine them.
How do you count the factors of a number?
Write the prime factorization, add 1 to each exponent and multiply. 48 = 2⁴ × 3¹ has (4 + 1)(1 + 1) = 10 factors, and 10¹⁵ = 2¹⁵ × 5¹⁵ has 16 × 16 = 256. The count is odd only for perfect squares, because their square root pairs with itself: 36 has 9 factors, with 6 × 6 in the middle.
What is a perfect number?
A number equal to the sum of its proper factors, the factors other than itself. 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are the first two; the next are 496, 8,128 and 33,550,336. Euclid showed that 2ᵖ⁻¹(2ᵖ − 1) is perfect whenever 2ᵖ − 1 is prime, and Euler proved every even perfect number has that form. Whether an odd perfect number exists is still unknown.
What are abundant and deficient numbers?
A number is abundant when its proper factors add up to more than it, and deficient when they add up to less. 12 is the smallest abundant number, since 1 + 2 + 3 + 4 + 6 = 16. Every prime is deficient, because its only proper factor is 1. Most small numbers are deficient; the smallest odd abundant number is 945, whose proper factors sum to 975.
What is the difference between factors and prime factors?
Factors are all the numbers that divide n; prime factors are the primes among them, and multiplying them with their repeats gives n. 48 has 10 factors but only two prime factors, 2 and 3, and its prime factorization is 2⁴ × 3. The prime factorization is unique, by the fundamental theorem of arithmetic, and every factor is a product of some of those primes.
Quelle est la précision de « Calculateur de diviseurs : tous les diviseurs d’un nombre » ?
La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 10. Par exemple, « 48 (default, Calculator Soup example) » est vérifié à l’aide de Calculator Soup factors calculator: the 10 factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; σ(48) = 124 by Python divisor enumeration.
D’où vient cette méthode ?
Hardy & Wright, An Introduction to the Theory of Numbers, §16.7 (the divisor functions d(n) and σ(n)) and §16.8 (perfect numbers); OEIS A000396: Perfect numbers; OEIS A005231: Odd abundant numbers; Wolfram MathWorld: Abundant number.