Split until only primes are left
A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11 and 13. A whole number above 1 that is not prime is composite, which means it can be written as two smaller numbers multiplied together: 24 = 4 × 6.
To break 24 into primes, split it into two factors, then split again every factor that is not prime. 4 is not prime, so split it: 4 = 2 × 2. 6 is not prime either: 6 = 2 × 3. Now every number at the end of a branch is prime, so the splitting stops.
Writing a number as primes multiplied together is called its prime factorization. The prime factorization of 24 is 2 × 2 × 2 × 3.
24 splits into 4 × 6. Then 4 splits into 2 × 2 and 6 into 2 × 3, and the branches end on the primes 2, 2, 2 and 3.
A different split, the same primes
Start again with a different split: 24 = 3 × 8. 3 is prime already, so that branch stops at once. 8 = 2 × 4, and 4 = 2 × 2. The branches end on 3, 2, 2 and 2: the same four primes as before, found in a different order.
Any first split gives them. 24 = 2 × 12, then 12 = 2 × 6 and 6 = 2 × 3, which ends on 2, 2, 2 and 3 once more.
Every whole number above 1 has exactly one prime factorization. However you split the number, the primes at the ends of the branches are the same. This fact is called the Fundamental Theorem of Arithmetic.
With 3 × 8 as the first split the tree has a different shape, but its branches still end on 2, 2, 2 and 3.
24 = 2 × 2 × 2 × 3 whichever split comes first: the primes at the ends of the branches are always the same
Start the tree with 2 × 12
Each first split grows a tree of a different shape. Every one of them ends on the same primes: 24 = 2 × 2 × 2 × 3.
A bigger number
For 180, start with an easy split: 180 = 10 × 18. Then 10 = 2 × 5 and 18 = 2 × 9, and 9 = 3 × 3. The prime factorization of 180 is 2 × 2 × 3 × 3 × 5.
Multiply the primes back together to check: 2 × 2 = 4, 4 × 3 × 3 = 36, and 36 × 5 = 180.
180 = 10 × 18, and the branches end on the primes 2, 5, 2, 3 and 3.
When no split comes to mind
Try the primes in order: divide by 2 for as long as you can, then by 3, then by 5, and so on. 90 is even, so 90 = 2 × 45. 45 is odd, so 2 does not divide it, but 3 does: 45 = 3 × 15. 15 = 3 × 5, and 5 is prime. So 90 = 2 × 3 × 3 × 5.
Go all the way down
The usual mistake is to stop too soon. 24 = 4 × 6 is true, but 4 and 6 are not prime, so it is not the prime factorization. Keep splitting until every number at the end of a branch is prime.
Splitting off a 1 does not help: 24 = 1 × 24 leaves 24 still to split, and 1 is not prime, because it has only one factor. It never appears in a prime factorization.