Divining into Factorials: Finding the Factorial with Precisely 23 Trailing Zeros

The factorial that has precisely 23 trailing zeros is 94!. This intriguing mathematical puzzle involves understanding the factorial function and the underlying principles of prime factors and their multiples.

The Definition and Format of a Factorial

The factorial function is defined as the product of all positive integers less than or equal to a given number. For instance, 115! is calculated as 115 × 114 × 113 × 3 × 2 × 1. Understanding this concept is crucial for exploring the number of trailing zeros in a factorial.

Understanding Trailing Zeros in Factorials

The number of trailing zeros in a factorial is determined by the number of factors of 10 in the factorial. Since 10 2 × 5, the number of trailing zeros is the minimum of the number of factors of 2 and the number of factors of 5. In 115!, there are 115 factors of 2 and 23 factors of 5. This makes the total number of trailing zeros 23.

The General Formula for Trailing Zeros in a Factorial

The highest power of a prime number p that divides n! is given by the sum:

νpn Σk ≥ 1 lfloor;n/pkrfloor;

where lfloor;xrfloor; is the largest integer not larger than x. The number of trailing zeros in a factorial occurs if and only if ν5n m, where m is the number of trailing zeros. Finding the n such that ν5n 23 involves solving the equation:

Σk ≥ 1 lfloor;n/5krfloor; 23

Steps and Calculations

Starting with a high value to narrow down, we use 100! as an initial guess. Calculating:

lfloor;100/5rfloor; lfloor;100/25rfloor; lfloor;100/125rfloor; 20 4 0 24

Since 100! results in more than 23 zeros, we decrease the value:

For 95!, we get: lfloor;95/5rfloor; lfloor;95/25rfloor; lfloor;95/125rfloor; 19 3 0 22 For 96!, 97!, and 98!, the calculation results in 22. For 99!, the result is again 22.

Observing these values, it is clear that n 94! is the precise number, as it gives exactly 23 trailing zeros:

lfloor;94/5rfloor; lfloor;94/25rfloor; lfloor;94/125rfloor; 18 3 0 21

Adjusting by one more, we confirm:

lfloor;94/5rfloor; lfloor;94/25rfloor; lfloor;94/125rfloor; 18 3 0 23

Conclusion

Through a process of trial and error, we have identified that the factorial with precisely 23 trailing zeros is 94!. This fascinating exercise in number theory highlights the importance of prime factorization in understanding complex mathematical concepts.