Understanding the Unit Digit Problem

The question asks for the unit digit of the product formed by multiplying all integers from 91 to 99. The expression is:

$ 91 \times 92 \times 93 \times \dots \times 99 $

Calculating the Unit Digit Rule

To determine the unit digit of a multiplication result, we only need to consider the unit digits of the numbers involved.

The unit digits for the numbers 91, 92, 93, ..., 99 are 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.

The unit digit of the overall product is equivalent to the unit digit of the product of these individual unit digits:

$ \text{UnitDigit}(91 \times 92 \times \dots \times 99) = \text{UnitDigit}(1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9) $

Key Insight: The Role of 5 and Even Numbers

Within the sequence of numbers from 91 to 99, we can identify critical factors:

• The number 95 ends in the digit 5.
• The sequence contains multiple even numbers, such as 92, 94, 96, and 98.

A fundamental property of multiplication is that any integer ending in 5, when multiplied by any even integer, always results in a product whose unit digit is 0.

Examples:

• $5 \times 2 = 10$ (Unit digit is 0)
• $5 \times 4 = 20$ (Unit digit is 0)
• $5 \times 6 = 30$ (Unit digit is 0)

Since the product $91 \times 92 \times \dots \times 99$ includes both 95 (a number ending in 5) and at least one even number, the unit digit of the entire product must be 0.

Final Determination

The unit digit calculation simplifies due to the presence of 5 and an even number:

$ \text{UnitDigit}(1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9) $

This product contains the factors 5 and 2 (among others). Their multiplication yields 10, which has a unit digit of 0.

$ \text{UnitDigit}( \dots \times 5 \times \dots \times 2 \times \dots ) = \text{UnitDigit}( \dots \times 10 \times \dots ) = 0 $

Therefore, the unit digit of the product $91 \times 92 \times 93 \times \dots \times 99$ is 0.