maths.free › Number Theory › Prime factorisation

Prime factorisation

Every integer is a product of primes in exactly one way.

Divide out the smallest prime again and again until only 1 is left; the primes you removed, with their multiplicities, are the unique prime factorisation. The factor tree below is drawn from those divisions. From it you can read the number of divisors and the gcd or lcm with any other number.

Δουλεμένο παράδειγμα · prime factorization of 360

Number-Theory · Prime-Factorization

Prime factorization of 360

360

Βήμα προς βήμα

  1. 360

    Factor 360 by dividing out the smallest prime that goes in, again and again.

  2. 360 = 2 \times 180

    2 is the smallest prime dividing 360.

  3. 180 = 2 \times 90

    2 is the smallest prime dividing 180.

  4. 90 = 2 \times 45

    2 is the smallest prime dividing 90.

  5. 45 = 3 \times 15

    3 is the smallest prime dividing 45.

  6. 15 = 3 \times 5

    3 is the smallest prime dividing 15. That leaves a prime, so we stop.

  7. 5 = 5 \times 1

    5 is the smallest prime dividing 5. That leaves a prime, so we stop.

  8. 360 = 2^{3} \times 3^{2} \times 5

    Collect equal primes as powers.

  9. d(360) = (3+1) \cdot (2+1) \cdot (1+1) = 24

    Bonus: the number of divisors is the product of (exponent + 1) over the prime powers.

Αποκάλυψέ την.
360 = 2^{3} \times 3^{2} \times 5

Τώρα εσύ. Επιλέξτε ένα πρόβλημα, ή πληκτρολογήστε ή σχεδιάστε το δικό σας.

Να δουλεύεις μόνος σου.

Ένας δωρεάν λογαριασμός προσθέτει σημειώσεις σε κάθε μάθημα, ένα αρχείο του τι έχετε τελειώσει, λύθηκαν προβλήματα σας σε ένα μέρος, και ένας δάσκαλος που μπορείτε να ρωτήσετε για αυτή τη σελίδα.

Υπογράψτε Σύνδεση

Σύμβολα που χρησιμοποιούνται εδώ

Χτυπήστε οποιοδήποτε σύμβολο για τον πλήρη ορισμό, μια εικόνα, και τι σημαίνει κάθε γράμμα σε αυτό.

Πώς να: Prime factorisation

  1. Factor 360 by dividing out the smallest prime that goes in, again and again.
  2. 2 is the smallest prime dividing 360.
  3. 2 is the smallest prime dividing 180.
  4. 2 is the smallest prime dividing 90.
  5. 3 is the smallest prime dividing 45.
  6. 3 is the smallest prime dividing 15. That leaves a prime, so we stop.
  7. 5 is the smallest prime dividing 5. That leaves a prime, so we stop.
  8. Collect equal primes as powers.

Ερωτήσεις που κάνουν οι άνθρωποι

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller-Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

Περισσότερα σε Number Theory

Prime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory