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.

Vinna dæmi · prime factorization of 360

Number-Theory · Prime-Factorization

Prime factorization of 360

360

Skref fyrir skref

  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.

Sýna svarið
360 = 2^{3} \times 3^{2} \times 5

Núna ūú Veldu vandamál, eða sláðu inn eða teikna eigin. Hvert skref, mynd, svarið falið þar til þú spyrð.

Haltu eigin vinnu þinni

Ókeypis reikningur bætir við athugasemdum við hverja kennslustund, skrá yfir það sem þú hefur lokið, leyst vandamál á einum stað og kennari sem þú getur spurt um þessa síðu.Stærðfræðin sjálf er opin öllum, skráður inn eða ekki.

Skráðu þig inn Innskráning

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Hvernig á að nota: 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.

Spurningar sem fólk spyr

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.

Meira í Number Theory

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