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A factorization tree is a rooted tree whose root is an integer $n>1$. Each nonleaf label is a composite number split into two factors greater than one whose product equals that label; the leaves are prime numbers. Multiplying the leaves recovers $n$. Many factorization trees can have the same root, but the fundamental theorem of arithmetic gives them the same leaf multiset.

Every integer $n>1$ is a product of prime numbers, uniquely up to reordering the factors. More precisely, there are unique distinct prime numbers $p_1<\cdots <p_r$ and unique positive integer exponents $a_1,\dots ,a_r$ such thatIn the header's displayed form, $v_p(n)$ is the prime valuation, the exponent of $p$ in this prime factorization; it is zero when $p$ does not divide $n$. Reordering or regrouping the prime factors does not give a different prime factorization.

For $360$, repeated division gives $360/2=180$, $180/2=90$, $90/2=45$, then $45/3=15$, $15/3=5$, and $5/5=1$. ThusIts nonzero prime valuations are $v_2(360)=3$, $v_3(360)=2$ and $v_5(360)=1$. Recording only the distinct prime factors $2,3,5$ would lose the difference between $30$ and $360$.

This turns divisibility into coordinatewise comparison of nonnegative exponents. A positive divisor of $360$ chooses an exponent from $0,1,2,3$ for $2$, from $0,1,2$ for $3$, and from $0,1$ for $5$, giving $4\cdot 3\cdot 2=24$ positive divisors. The greatest common divisor takes the smaller exponent at each prime number; the least common multiple takes the larger. The theorem is a structural description, not an assertion that finding the factors of every large integer is computationally easy.

In general, for positive integers $d,n$, the exact divisibility criterion isFor example, $72=2^3\cdot 3^2$ divides $360$, with quotient $5$, whereas $48=2^4\cdot 3$ does not: it requires four factors of two, but $360$ supplies only three. Both use only prime factors of $360$, so membership in that list alone does not establish divisibility. Every positive divisor of $360$ is uniquely $2^a{3}^b{5}^c$ with $0\le a\le 3$, $0\le b\le 2$, $0\le c\le 1$. Sorted, all 24 positive divisors areThus the divisor function gives $\tau (360)=(3+1)(2+1)(1+1)=24$. These include divisors that are composite numbers, as well as the prime factors and one.

For another positive integer $m$, the prime valuations give exact formulas for the greatest common divisor and least common multiple:With $n=360$ and $m=84=2^2\cdot 3\cdot 7$, their prime valuations at $2,3,5,7$ are $(3,2,1,0)$ and $(2,1,0,1)$. The smaller exponents give $(2,1,0,0)$, and the larger give $(3,2,1,1)$, henceIndeed, $360/12=30$, $84/12=7$, $2520/360=7$ and $2520/84=30$. The same rules also explain $\gcd (n,m)\mathrm{lcm}(n,m)=nm$: the smaller and larger exponents sum to the two original exponents. Here $12\cdot 2520=360\cdot 84=30240$.

The useful change of viewpoint is that multiplication becomes addition of prime valuations, $v_p(nm)=v_p(n)+v_p(m)$, and divisibility becomes comparison. Many different-looking calculations therefore share one representation. The prime valuations are coordinates for multiplication, rather than digits in positional notation: they do not turn ordinary addition of integers into coordinatewise addition.

Different intermediate splits lead to the same prime ingredients. Starting with $360=36\cdot 10$ gives $36=6\cdot 6$ and $10=2\cdot 5$; starting with $360=8\cdot 45$ gives $8=2\cdot 4$ and $45=5\cdot 9$. Continuing to split the composite numbers produces the same multiset of prime factors: three twos, two threes, and one five. The factorization tree records choices in the calculation, while the prime factorization records the invariant result.

The proof outline explains the uniqueness. Existence follows by splitting a composite number into smaller factors until only prime numbers remain. For uniqueness, Euclid's lemma says that a prime factor of one product must divide one factor of any other prime product equal to it. That factor must be the same prime number; cancel it and repeat. The crucial divisibility step appears in Book VII, Proposition 30 of Euclid's Elements. ^[ref]

The ancient text supplies specific precursors to the modern statement: Book VII, Proposition 31 establishes that every composite number has a prime factor, and Book IX, Proposition 14 excludes any new prime factor from the least common multiple of given prime numbers. These are the propositions being cited, rather than the complete modern exponent formulation of the fundamental theorem of arithmetic. ^{[ref] [ref]}

Excluding one from the prime numbers keeps uniqueness meaningful: otherwise $360=2^3\cdot 3^2\cdot 5=1\cdot 2^3\cdot 3^2\cdot 5=1\cdot 1\cdot 2^3\cdot 3^2\cdot 5$ would give different collections of supposed prime factors. For $n=1$, the natural convention is the empty product, equal to one, with no prime factors and all prime valuations zero.

The total number of prime factors $\Omega (n)$ counts every occurrence of a prime factor in the prime factorization of a positive integer $n$. Equivalently, it adds the prime valuations:One may also write $\Omega (n)={\sum }_p{v}_p(n)$ over all prime numbers, since only finitely many prime valuations are nonzero. The empty sum gives $\Omega (1)=0$. By contrast, the prime omega function has $\omega (n)=r$ and counts distinct prime factors. ^[ref]

Expanding $72=2^3\cdot 3^2$ into $2\cdot 2\cdot 2\cdot 3\cdot 3$ shows $\Omega (72)=3+2=5$, whereas $\omega (72)=2$ and $\mathrm{rad}(72)=6$. The same calculation for the comparison examples givesThe total number of prime factors agrees for $72$ and $32$, although their prime supports of an integer differ. Meanwhile $72$ and $36$ have the same radical of an integer and prime omega function, but different values of the total number of prime factors. These examples show exactly which information each arithmetic function forgets.

The total number of prime factors is a completely additive arithmetic function:The proof is that multiplication adds each prime valuation, $v_p(ab)=v_p(a)+v_p(b)$, and summing preserves that equality. No coprimality condition is needed. For example, $12=2^2\cdot 3$ and $18=2\cdot 3^2$ both have total number of prime factors three; their product $216=2^3\cdot 3^3$ has six. By comparison, the prime omega function is only an additive arithmetic function: here $\omega (216)=2\ne 4=\omega (12)+\omega (18)$.

There is a useful way to see the total number of prime factors as a count of elementary building blocks: any expression of $n$ as a product of prime numbers has exactly $\Omega (n)$ factors, by the fundamental theorem of arithmetic. Equivalently, every factorization tree for $n>1$ has $\Omega (n)$ leaves. Repeated prime factors increase this count without introducing new kinds of prime number. The difference is the existing excess prime-factor multiplicity:Every summand is nonnegative, so $\Omega (n)=\omega (n)$ if and only if $n$ is a squarefree integer, including $n=1$. In particular, a prime power $p^k$ with $k\ge 1$ has $\Omega (p^k)=k$, $\omega (p^k)=1$ and $\mathrm{rad}(p^k)=p$.

Historically, Hardy and Ramanujan's 1917 paper explicitly separated the prime omega function from the total number of prime factors, writing them as $f(n)$ and $F(n)$, respectively. Their example $2^3\cdot 3^2\cdot 5$ has counts three and six. That distinction let them compare two measures of how an integer is built from prime factors. ^[ref]