What is the unit's digit in the product? $81 \times 82 \times 83 \times \dots \times 89$

### Concept & Logic The unit digit of a large multiplication expression depends entirely on the product of the individual unit digits. If any combination of those unit digits produces a zero, the unit digit for the entire massive product will definitively be zero. ### Step-by-Step Solution - **Calculation / Deduction:** The given sequence is $81 \times 82 \times 83 \times 84 \times 85 \times \dots \times 89$. - Extract only the unit digits to evaluate: $1 \times 2 \times 3 \times 4 \times 5 \times \dots \times 9$. - Notice that the sequence contains a 2 (from 82) and a 5 (from 85). - When multiplied, $2 \times 5 = 10$, which ends in a 0. - Any integer multiplied by a number ending in 0 will result in a product that also ends in 0. ### Exam Strategy & Shortcut Rule of thumb: Whenever you are asked for the unit digit of a continuous sequence of 5 or more consecutive numbers, the answer is always **0**. This is because any sequence of 5 numbers guarantees the presence of at least one even number (ends in 2, 4, 6, 8) and one multiple of 5, which combine to create a trailing zero. ### Common Pitfall Students sometimes start manually multiplying the sequence ($1 \times 2 = 2$, $2 \times 3 = 6$, $6 \times 4 = 24$, etc.) which wastes valuable time and introduces the possibility of arithmetic mistakes before reaching the 5. Always scan the series for a 2 and a 5 first. ### Final Answer Therefore, the correct answer is **0**.