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Mathematics · pattern A, Numeric fields

Prime factorisation and every divisor.

Enter a number for its prime factorisation, the complete list of divisors, how many there are and their sum.

Inputs

The formula used

every integer above 1 factorises uniquely into primes

The divisor count follows from the exponents: add one to each and multiply. 12 = 2²×3 has (2+1)(1+1) = 6 divisors.

Prime factorisation

2 × 2 × 2 × 3 × 3 × 5

360 factorises, so it is composite.

Prime factorisation
2 × 2 × 2 × 3 × 3 × 5
All divisors
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72
Number of divisors
24
Sum of divisors
1170
Perfect square?
no

Worked steps

  1. 1divide by 2

    trial division by successive primes until the quotient is 1

  2. 2divide by 2

    trial division by successive primes until the quotient is 1

  3. 3divide by 2

    trial division by successive primes until the quotient is 1

  4. 4divide by 3

    trial division by successive primes until the quotient is 1

  5. 5divide by 3

    trial division by successive primes until the quotient is 1

  6. 6divide by 5

    trial division by successive primes until the quotient is 1

Unique factorisation is the fundamental theorem.

Every integer above one has exactly one prime factorisation, up to the order of the factors. That uniqueness is what makes factorisation a canonical form, and why the divisor-count formula and the GCF-from-factorisation method both work exactly.

Questions about factor

How do I count divisors without listing them?
Add one to each exponent and multiply. For 360 = 2³×3²×5 that is 4 × 3 × 2 = 24.
What is a perfect number?
One whose proper divisors sum to itself, such as 6 and 28. The divisor sum above lets you check.
Why is factorisation hard for large numbers?
No efficient general algorithm is known, which is what public-key cryptography relies on.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.