Math Puzzle: What Is the Last Digit of the Sum of Factorials from 1 to 2026?

This week's mathematical puzzle challenges readers to determine the final digit of a sum involving factorials: 1! + 2! + 3! + ... + 2026!.

The problem, presented without the aid of calculators or AI, requires understanding the properties of factorials. A factorial (denoted by an exclamation mark) is the product of all positive integers up to a given number. For instance, 3! equals 1 * 2 * 3, which is 6.

The key to solving this puzzle lies in observing the pattern of the last digits of factorials. While 1! = 1, 2! = 2, 3! = 6, and 4! = 24, all subsequent factorials (5! and higher) will always end in zero. This is because each of these factorials includes the numbers 2 and 5 in their product, which multiply to 10. Any number multiplied by 10 ends in zero.

Consequently, when calculating the last digit of the sum 1! + 2! + 3! + ... + 2026!, all terms from 5! onwards can be disregarded as they contribute only a zero to the final digit. The sum of the last digits is therefore determined solely by the first four terms: 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. The final digit of this sum is 3.