# Çarpan hesaplayıcı: bir sayının tüm bölenleri

> 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.

Etkileşimli sürüm: https://www.calcopenly.com/tr/math/factor-calculator
Konu: Matematik hesaplayıcıları

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.

## Girdiler

- **Whole number**: Up to 10¹⁸. Expressions work too, such as 2^40 or 10^15.

## Sonuçlar

- Çarpanlar — ana sonuç
- Number of factors
- Factor pairs
- Prime factorization
- Sum of all factors σ(n)
- Sum of proper factors
- Perfect, abundant or deficient
- Prime or composite

## Formül

$$
n = \prod p_i^{e_i} \quad d(n) = \prod (e_i + 1) \quad \sigma(n) = \prod \frac{p_i^{e_i+1} - 1}{p_i - 1} \quad s(n) = \sigma(n) - n
$$

## Çözümlü örnekler

### 48 (default, Calculator Soup example)

- Whole number: 48
- **Çarpanlar: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48**
- **Number of factors: 10**
- **Factor pairs: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8**
- **Prime factorization: 2⁴ × 3**
- **Sum of all factors σ(n): 124**
- **Sum of proper factors: 76**
- **Perfect, abundant or deficient: Abundant: the proper factors add up to 76, more than 48**
- **Prime or composite: Composite**
- Doğrulama kaynağı: 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

### 36, a perfect square (Calculator Soup example)

- Whole number: 36
- **Çarpanlar: 1, 2, 3, 4, 6, 9, 12, 18, 36**
- **Number of factors: 9**
- **Factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6**
- **Sum of all factors σ(n): 91**
- Doğrulama kaynağı: Calculator Soup factors calculator: factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36; σ(36) = 91 by Python

### 120 (calculator.net factor tree)

- Whole number: 120
- **Prime factorization: 2³ × 3 × 5**
- **Number of factors: 16**
- **Sum of all factors σ(n): 360**
- **Sum of proper factors: 240**
- Doğrulama kaynağı: calculator.net factor calculator: 120 = 2 × 2 × 2 × 3 × 5; 16 divisors summing to 360 by Python enumeration

### 28 is perfect

- Whole number: 28
- **Çarpanlar: 1, 2, 4, 7, 14, 28**
- **Sum of proper factors: 28**
- **Perfect, abundant or deficient: Perfect: the proper factors add up to exactly 28**
- Doğrulama kaynağı: OEIS A000396 (perfect numbers: 6, 28, 496, 8128, …); 1 + 2 + 4 + 7 + 14 = 28

### 945, the smallest odd abundant number

- Whole number: 945
- **Number of factors: 16**
- **Sum of proper factors: 975**
- **Perfect, abundant or deficient: Abundant: the proper factors add up to 975, more than 945**
- Doğrulama kaynağı: OEIS A005231 (odd abundant numbers start 945, 1575, 2205); σ(945) = 1920 by Python

### 8,589,869,056, the 6th perfect number

- Whole number: 8589869056
- **Prime factorization: 2¹⁶ × 131071**
- **Number of factors: 34**
- **Sum of proper factors: 8,589,869,056**
- Doğrulama kaynağı: OEIS A000396: 6th perfect number 8589869056 = 2¹⁶ × (2¹⁷ − 1); 131071 prime (Mersenne prime M17)

## Sorular

### 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.

### “Çarpan hesaplayıcı: bir sayının tüm bölenleri” ne kadar doğru sonuç verir?

Doğruluk, girdilerinize ve yöntemin varsayımlarına bağlıdır. Ondalık aritmetik 50 anlamlı basamak kullanır; ancak tahminler, sayısal yöntemler ve kaynak veriler daha az hassas olabilir. Gösterilen değerin yuvarlanması bu sınırları ortadan kaldırmaz. Bağımsız kaynaklarla doğrulanan çözümlü örnek sayısı: 10. Örneğin “48 (default, Calculator Soup example)”, 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 ile karşılaştırılarak doğrulanır.

### Yöntemin kaynağı nedir?

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.

## Kaynaklar

- 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](https://oeis.org/A000396)
- [OEIS A005231: Odd abundant numbers](https://oeis.org/A005231)
- [Wolfram MathWorld: Abundant number](https://mathworld.wolfram.com/AbundantNumber.html)