A balanced factor tree terminating in color-coded prime leavesUpdates with every input
Math & Statistics
Prime Factorization Solver Calculator
Factor a positive integer into unique prime powers and derive divisor count, divisor sum, Euler's totient, radical, square-free status, perfect-power evidence, and a live factor tree.
Prime factorization-
Distinct prime factors-
Prime factors with multiplicity-
Number of divisors-
Sum of positive divisors-
Euler totient φ(n)-
Prime radical rad(n)-
Structure classification-
MULTIPLICATIVE ANATOMY
A balanced factor tree terminating in color-coded prime leaves
Composite nodes split recursively; repeated prime leaves share a color and feed an exponent summary rail.
PRIME-POWER LEDGER
How each prime power contributes to arithmetic functions
Each row isolates the local contribution to divisor count, divisor sum, totient, and radical.
| Prime p | Exponent a | Prime power p^a | τ contribution a+1 | σ contribution | φ contribution |
|---|
INTEGER SETUP
Enter an exact positive integer, not a rounded measurement
- Use an integer of at least two.
- Keep identifiers with leading zeros out of arithmetic interpretation.
- Read exponents as repeated prime multiplicities.
- Use the ledger for divisor-function calculations.
- Verify the product of prime powers reconstructs the input.
UNIQUE DECOMPOSITION
Order can change, but the prime multiset cannot
Different factor trees may split a composite number in different orders. After all leaves are prime, their multiset is unique.
The factorization is therefore an exact certificate: multiplying the displayed prime powers must return the original integer.
FUNDAMENTAL THEOREM
Reduce the integer to one unique multiset of primes
Trial division removes the smallest available prime repeatedly. The resulting prime exponents determine several multiplicative arithmetic functions without enumerating every divisor.
Detailed calculation process and general formulas
n = product p_i^a_itau(n) = product(a_i+1)sigma(n) = product[(p_i^(a_i+1)-1)/(p_i-1)]phi(n) = n product(1-1/p_i)rad(n) = product p_iSymbols, meanings, and units
- p_i
- distinct prime divisorinteger
- a_i
- exponent of prime p_icount
- tau
- number of positive divisorsdivisors
- sigma
- sum of positive divisorsinteger
- rad
- product of distinct prime divisorsinteger
NUMBER-THEORY PROFILE
One factorization answers several structural questions
Prime exponents encode much more than a product.
Divisor abundance
-Exponent choices create every positive divisor.
Coprime residues
-Euler's product removes multiples of each distinct prime.
Repeated structure
-Exponent gcd and square-free status distinguish perfect powers and repeated factors.
Decision takeaway: Use the prime-power ledger as the reusable certificate, not the visual split order.
Applied decisions
Prime-factorization applications
Fraction reduction and LCM
Several denominators are decomposed into prime powers.
What the result clarifies: Minimum and maximum exponents recover gcd and lcm systematically.
Divisor enumeration planning
A combinatorial routine needs the count of divisors before generating them.
What the result clarifies: The product of exponent-plus-one terms gives the exact count.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
The browser implementation uses exact integer arithmetic within the displayed input range. It is intended for instructional and moderate-size arithmetic, not cryptographic factorization or primality certification of very large integers.
Prime Factorization Solver Calculator | Factor Tree, Divisors, Totient FAQ
Can two correct factor trees look different?
Yes. Their split order can differ, but their final prime leaves and exponents must match.
Why does exponent a contribute a+1 divisors?
A divisor can use p to any exponent from zero through a.
Is 1 prime?
No. It has no prime factorization and is excluded from the input range.