Enter a positive integer to list all proper divisors, count, sum and product
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Proper Divisor Count
Sum of Proper Divisors
Product of Divisors
Step-by-Step Calculation
Divisor Count & Sum Formulas
If n = p₁a₁ × p₂a₂ × … × pkak
Divisor count d = (a₁+1)(a₂+1)…(ak+1)
Divisor sum σ(n) = [(p₁a₁+1-1)/(p₁-1)] × … × [(pkak+1-1)/(pk-1)]
Proper divisor calculation is fundamental in number theory, widely used for perfect number checking, amicable number analysis and factorization tasks.
⚠Note: Proper divisors exclude the number itself, while positive divisors include 1 and the number. A perfect number has a proper divisor sum equal to itself. Numbers with sum > n are abundant, sum < n are deficient.
What Are Proper Divisors?
A proper divisor of a positive integer n is a positive integer less than n that divides n exactly (excluding n itself). It is essential for perfect number testing, amicable number research and divisor analysis.
Proper Divisor Definition
A proper divisor is a positive integer less than n that divides n without remainder, excluding n itself. Proper divisors of 12: 1, 2, 3, 4, 6.
Divisor Count Theorem
If n = p₁^a₁×...×p_k^a_k, divisor count d=(a₁+1)(a₂+1)...(a_k+1). For 12=2²×3¹, count=(2+1)(1+1)=6.
Divisor Sum Function
The sum of all positive divisors σ(n) uses prime factorization: σ(n) = product of (p^(a+1)-1)/(p-1) for each prime power.
Number Classification
Sum = n → Perfect Number; Sum > n → Abundant Number; Sum < n → Deficient Number. Used for perfect/amicable number detection.
💡 Example: Find proper divisors of 12. 12=2²×3¹, total divisors=6. Proper divisors: 1,2,3,4,6 (count=5). Sum=1+2+3+4+6=16. 12 < 16 → 12 is an abundant number.
Applications
Perfect NumbersAmicable NumbersFactorizationMath ContestsNumber Theory
Frequently Asked Questions
What is a proper divisor?▼
A proper divisor of a positive integer n is a positive divisor less than n (excluding n itself). For example, proper divisors of 12 are 1, 2, 3, 4, 6.
What is the divisor count formula?▼
If n = p₁^a₁ × p₂^a₂ × … × p_k^a_k, the number of positive divisors is (a₁+1)(a₂+1)…(a_k+1). For 12=2²×3¹, count=6.
What is a perfect number?▼
A perfect number is a positive integer where the sum of all proper divisors equals the number itself. The total divisor sum equals 2n. Example: 6 is perfect (1+2+3=6).
What is the divisor product rule?▼
For any integer n, the product of all positive divisors equals n^(d/2), where d is the total number of divisors. Divisors form pairs multiplying to n.
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