Mathyessy
Home › Topics › Number Theory › Trailing zeros in factorials

Trailing zeros in factorials practice

Each trailing zero comes from a factor of 10 = 2 × 5. In a factorial, the 5s are the bottleneck.

Practice this in Mathyessy →

How it works

A number ends in as many zeros as the number of times 10 = 2 × 5 divides it. In n! there are always more 2s than 5s, so just count the 5s:

⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

The first term counts the multiples of 5, the second adds one more for each multiple of 25 (which contributes two 5s), and so on. Stop when the power of 5 is bigger than n.

Going backwards ("smallest n with 10 zeros"), note that the count only increases at multiples of 5, and it jumps by 2 at multiples of 25. Some counts are skipped entirely: no factorial ends in exactly 5 zeros.

Trailing Zeros In Factorials practice problems with solutions

These problems come from Mathyessy's question bank. Try each one before opening the solution.

  1. How many trailing zeros does 25 factorial have?

    Show answer and solution

    Answer: 6

    ⌊25/5⌋ + ⌊25/25⌋ = 5 + 1 = 6.

  2. How many trailing zeros does 100 factorial have?

    Show answer and solution

    Answer: 24

    ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24. (125 is bigger than 100, so stop.)

  3. How many trailing zeros does 1000 factorial have?

    Show answer and solution

    Answer: 249

    ⌊1000/5⌋ + ⌊1000/25⌋ + ⌊1000/125⌋ + ⌊1000/625⌋ = 200 + 40 + 8 + 1 = 249.

  4. What is the smallest n so that n factorial has 10 trailing zeros?

    Show answer and solution

    Answer: 45

    40! has 8 + 1 = 9 zeros, and 41! to 44! add no new 5s. 45! adds one more 5, giving 10 zeros, so n = 45.

Practice this in Mathyessy →Free, no account needed. Opens the Trailing Zeros In Factorials set with instant feedback.

Keep going

Related topics

Related practice sets in the app

All Number Theory topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice