# How many zeroes end 100!?: visual puzzle | SwaVid

The number 100! equals 100 × 99 × 98 × … × 1. How many consecutive zeroes are at its end? See the complete step-by-step solution.

Canonical: https://www.swavid.com/puzzles/trailing-zeroes

Source: https://www.swavid.com/puzzles/trailing-zeroes

# How many zeroes end 100!?

## 24 zeroes.

### Count multiples of 5

### Count multiples of 25

### Add them

SwaVid visual puzzle

The number 100! equals 100 × 99 × 98 × … × 1. How many consecutive zeroes are at its end?

Detailed solution

Each trailing zero comes from a factor of 10 = 2 × 5. There are far more 2s than 5s, so count the 5s.

Every fifth number contributes at least one factor of 5.

25, 50, 75 and 100 contribute a second factor of 5.

125 is larger than 100, so there are no triple fives to add.

Answering 20 forgets that 25, 50, 75 and 100 each hide two fives.

- 1 Count multiples of 5 Every fifth number contributes at least one factor of 5. 100 ÷ 5 = 20
- 2 Count multiples of 25 25, 50, 75 and 100 contribute a second factor of 5. 100 ÷ 25 = 4
- 3 Add them 125 is larger than 100, so there are no triple fives to add. 20 + 4 = 24

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