One Tree per Number

A factor tree can start with any pair. Take 48. One tree splits it as 6 × 8, then 6 into 2 × 3 and 8 into 2 × 4, then 4 into 2 × 2. Another tree starts with 4 × 12. The branches differ, but both trees end with the same leaves: four 2s and one 3, 2^4 × 3.

That always happens: every whole number above 1 has exactly one prime factorization, whichever tree you draw. Only the order of the leaves can change, and the order of a product never matters. So start with whatever split you spot first; you cannot end up with different primes.

This is why primes are called the building blocks of the whole numbers. Each number is built from its own primes, and no other set of primes multiplies to it. If two people factor the same number and get different primes, one of them has made a slip.