Factorials & Power Cycles

This topic deals with the properties of factorials and the repeating patterns of unit digits in powers.

Factorial (!): The factorial of a non-negative integer 'n', denoted by \( n! \), is the product of all positive integers less than or equal to n. Example: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Note that \( 0! = 1 \).
Power Cycle (Cyclicity): This refers to the repeating pattern of the last digit (unit digit) of a number when it is raised to successive powers. For example, the unit digits of powers of 2 are 2, 4, 8, 6, 2, 4, ... The pattern '2, 4, 8, 6' repeats, so its cyclicity is 4.

Highest Power of a Prime 'p' in n!: To find the highest power of a prime number 'p' that can divide \( n! \), use the formula: \( \lfloor \frac{n}{p} \rfloor + \lfloor \frac{n}{p^2} \rfloor + \lfloor \frac{n}{p^3} \rfloor + ... \) where \( \lfloor x \rfloor \) is the greatest integer function.
Number of Trailing Zeros in n!: This is determined by the number of times 10 is a factor, which is the same as the highest power of 5 in \( n! \) (since the prime factorization of 10 is 2x5, and there are always more factors of 2 than 5). The formula is: \( \lfloor \frac{n}{5} \rfloor + \lfloor \frac{n}{25} \rfloor + \lfloor \frac{n}{125} \rfloor + ... \)
Unit Digit Cyclicity Chart:
- Cycle of 1: 0, 1, 5, 6 (unit digit is always the number itself)
- Cycle of 2: 4, 9 (4¹, 4² -> 4, 6; 9¹, 9² -> 9, 1)
- Cycle of 4: 2, 3, 7, 8

To find the unit digit of \( x^y \), find the remainder when 'y' is divided by the cyclicity of 'x'. If the remainder is 'r', the unit digit is the same as the unit digit of \( x^r \). If the remainder is 0, the unit digit is the same as the last digit in the cycle of x.
The number of trailing zeros in an expression like \( a! \times b! \times c! \) is the sum of the trailing zeros in each factorial.

Questions on this topic are about recognizing patterns, not performing heavy calculations.
For unit digit questions, you only ever need to focus on the unit digits of the base numbers.
Practice finding the highest power of both prime and composite numbers in a factorial to cover all question types.