Factor calculator: every factor and factor pair of a whole number up to 10¹⁸, its prime factorization, divisor count and sum, and perfect or abundant.
更新日 検証済みの例：10
48 = 2⁴ × 3 has 10 factors adding up to 124. Without 48 itself they add up to 76, so 48 is abundant.
| Factor d | Partner n ÷ d | d is |
|---|---|---|
| 1 | 48 | unit |
| 2 | 24 | prime |
| 3 | 16 | prime |
| 4 | 12 | composite |
| 6 | 8 | composite |
Trial division by primes below 10,000, then Pollard's rho method for any large cofactor.
Each factor chooses an exponent for every prime: 0 to 4 for 2, 0 to 1 for 3.
Only divisors up to √48 ≈ 6 need testing; each one brings its partner.
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.
照合元：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
照合元：Calculator Soup factors calculator: factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36; σ(36) = 91 by Python
照合元：calculator.net factor calculator: 120 = 2 × 2 × 2 × 3 × 5; 16 divisors summing to 360 by Python enumeration
照合元：OEIS A000396 (perfect numbers: 6, 28, 496, 8128, …); 1 + 2 + 4 + 7 + 14 = 28
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.
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.
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.
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.
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.
精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：10。 例えば、「48 (default, Calculator Soup example)」は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と照合しています。
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.
この計算機には、独立した出典の解答を使った計算例が 10 件あります。テストに組み込まれており、ここでも実行できます。
Prime factorization of any whole number up to 60 digits, with a factor tree, a prime or composite verdict, its divisors and the nearest primes.
Greatest common divisor (GCD, GCF or HCF) and least common multiple (LCM) of two or more whole numbers, with Euclid's algorithm step by step.
Square root calculator: the root to up to 1,000 decimal places, simplest radical form (√72 = 6√2), a perfect-square check and cube or nth roots.
Long division calculator that shows every step: quotient and remainder, the decimal quotient with its repeating block marked, and the fraction.
Calculate a mod n under the floored, truncated and Euclidean conventions, modular powers a^b mod m of large numbers, and modular inverses, with steps.