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# Prime factorisation

Every positive integer is the product of prime numbers. The list of prime factors of a number is called its prime factorisation. The prime factorisation of every number is unique.

$$6 = \Tblue{2}\times \Tgreen{3},\qquad 12 = \Tblue{2}\times\Tblue{2}\times \Tgreen{3},\qquad 1105 = \Tblue{5}\times \Tgreen{13}\times \Tviolet{17}.$$

The number of times that a prime factor appears is its multiplicity.

$\Tblue{2}$ has multiplicity $\Tred{3}$ in the factorisation of $24 = \Tblue{2}^\Tred{3} \times \Tgreen{3}$.

If the prime factorisation of a number is known, it is easy to find all the factors of that number. The factors are all possible products of the prime factors.

We have the prime factorisation $$24 = \Tblue{2}^\Tred{3} \times \Tgreen{3}.$$ So all the factors of $24$ are : $1$, $\Tblue{2}$, $\Tblue{2}^\Tred{2} = 4$, $\Tblue{2}^\Tred{3} = 8$, $\Tgreen{3}$, $\Tblue{2}\times\Tgreen{3} = 6$, $\Tblue{2}^\Tred{2} \times\Tgreen{3} = 12$ and $\Tblue{2}^\Tred{3} \times\Tgreen{3} = 24$.

The prime factorisation of a number can be found using a tree method. Find the highest