What is Prime Factorisation?

Prime factorisation allows us to write any number as a product of prime factors. For example, consider the number 360. Let us write this number as follows:

\[\begin{array}{l}360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\\\;\;\;\;\;\, = {2^3} \times {3^2} \times {5^1}\end{array}\]

Basically, we have expressed 360 as a product of prime numbers. Let us take some other (composite) numbers and express them as products of prime numbers:

\[\begin{array}{l}\;\,80 = 2 \times 2 \times 2 \times 2 \times 5\\\;\;\;\;\;\,\,\, = {2^4} \times {5^1}\\144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3\\\;\;\;\;\;\,\,\, = {2^4} \times {3^2}\\600 = 2 \times 2 \times 2 \times 3 \times 3 \times 5\\\;\;\;\;\;\,\,\,= {2^3} \times {3^2} \times {5^1}\end{array}\]

Expressing a number this way (as a product of primes) is called the prime factorization of that number. Note that the prime factorization of a prime number is trivial, since the only non-unity divisor of any prime number is that number itself.

Can every number be prime-factorised in a unique way?

Two questions now arise:

  1. Can every composite number be expressed as a product of primes, that is, can every composite number be prime factorized?

  2. If we prime factorize a composite number, is that factorization unique? That is, if we express a composite number as a product of primes, can it be expressed as a product of some other set of primes?

These questions are answered by one of the most important results in Mathematics, which we discuss next.

Learn math from the experts and clarify doubts instantly

  • Instant doubt clearing (live one on one)
  • Learn from India’s best math teachers
  • Completely personalized curriculum