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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.

ምሳሌ · prime factorization of 360

Number-Theory · Prime-Factorization

Prime factorization of 360

360

እርምጃ በደረጃ

  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.

መልሱን አሳይ
360 = 2^{3} \times 3^{2} \times 5

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

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ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: 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.

ጥያቄዎች

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.

በ Number Theory

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