The numbers $1, 2, 3, 4, ........., 1000$ are multiplied together. The number of zeros at the end (on the right) of the product must be

### Concept & Formula The product of consecutive integers from $1$ to $n$ is represented as $n!$ (n factorial). Trailing zeros in any product are created exclusively by the multiplication of pairs of prime factors $2$ and $5$ (since $2 \times 5 = 10$). In any factorial, the prime factor $2$ appears far more frequently than the prime factor $5$. Therefore, the number of trailing zeros is entirely limited by, and equal to, the total count of $5$'s in the prime factorization of $n!$. We use **Legendre's Formula** to count the highest power of a prime number $p$ dividing $n!$: $$ \text{Power of } p = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots $$ ### Step-by-Step Solution * **Given:** The product is $1 \times 2 \times 3 \times \dots \times 1000 = 1000!$ We need to find the highest power of $5$ that divides $1000!$. * **Calculation / Deduction:** Apply the successive division method (Legendre's formula) using $n = 1000$ and $p = 5$. * First, divide $1000$ by $5$: $$ \lfloor 1000 / 5 \rfloor = 200 $$ * Next, divide the quotient ($200$) by $5$ (which is equivalent to dividing $1000$ by $25$): $$ \lfloor 200 / 5 \rfloor = 40 $$ * Next, divide the new quotient ($40$) by $5$ (equivalent to dividing $1000$ by $125$): $$ \lfloor 40 / 5 \rfloor = 8 $$ * Finally, divide $8$ by $5$ (equivalent to dividing $1000$ by $625$): $$ \lfloor 8 / 5 \rfloor = 1 $$ (We only take the integer part) * Stop here because dividing $1$ by $5$ yields $0$. Add all the integer quotients together to find the total number of $5$'s: $$ \text{Total zeros} = 200 + 40 + 8 + 1 = 249 $$ ### Exam Strategy & Shortcut **Successive Division Ladder:** You do not need to write out the formal powers of $5$. Simply set up a division ladder. Divide $1000$ by $5$ to get $200$. Immediately divide $200$ by $5$ to get $40$. Divide $40$ by $5$ to get $8$. Divide $8$ by $5$ to get $1$. Sum the results vertically: $200 + 40 + 8 + 1 = 249$. This takes under 10 seconds. ### Common Pitfall A common error is stopping after the first division (yielding $200$) and selecting option (b). Students forget that multiples of higher powers of $5$ (like $25$, $125$, and $625$) contribute *extra* $5$'s to the factorization, which must be accounted for. ### Final Answer **Therefore, the correct answer is 249.**