Factorial Properties
Several factorial facts and the highest-power-of-a-composite technique recur in CAT. For a composite divisor, split it into prime powers, find each prime’s exponent in n! by Legendre, then take the limiting one. For 6 = 2 × 3 the answer is min(power of 2, power of 3); since 3 is always scarcer than 2, the power of 3 wins. For a prime square like 9 = 3², first find the power of 3, then take ⌊that/2⌋. Other staples: for n ≥ 5, n! always ends in zero, so the units digit of any sum like 5! + 6! + … is fixed by the small terms; n! + 1 and consecutive factorials show up in remainder questions; and ratios n!/(n−r)! collapse to a short product, which is how factorials feed permutations. Knowing 1! through 7! by heart (up to 5040) saves real time.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Core definitions & identities
| Factorial | n! = 1 × 2 × 3 × … × n |
|---|---|
| Empty product | 0! = 1! = 1 |
| Recurrence | n! = n × (n − 1)! |
| Ratio of factorials | n! / (n − r)! = n(n−1)…(n−r+1) |
| Combination | ⁿCᵣ = n! / [r!(n − r)!] |
Power & zero-counting tools
| Highest power of prime p in n! (Legendre) | ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … |
|---|---|
| Trailing zeros of n! | ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … |
| Power of composite (e.g. 6 = 2×3) | min of the prime-factor powers in n! |
| Power of pᵏ (e.g. 9 = 3²) | ⌊ (power of p in n!) / k ⌋ |
| Quick count of multiples of m up to n | ⌊n/m⌋ |