The number of consecutive zeros in $2^3 imes 3^4 imes 5^4 imes 7$ is equal to :

Question

The number of consecutive zeros in $2^3 	imes 3^4 	imes 5^4 	imes 7$ is equal to :

Accepted Answer

To determine the number of consecutive zeros at the end of the number $2^3 \times 3^4 \times 5^4 \times 7$, we need to consider the factors of 10 in the number. Each factor of 10 is composed of a factor of 2 and a factor of 5. Therefore, the number of consecutive zeros is determined by the minimum number of pairs of 2s and 5s in the prime factorization of the number.

Let's break down the prime factorization:

- $2^3$ provides 3 factors of 2.
- $3^4$ provides no factors of 2 or 5.
- $5^4$ provides 4 factors of 5.
- $7$ provides no factors of 2 or 5.

To form a factor of 10, we need one 2 and one 5. The limiting factor here is the number of 2s, as we have only 3 factors of 2 available. Therefore, we can form 3 pairs of 2s and 5s, resulting in 3 factors of 10.

Thus, the number of consecutive zeros at the end of the number is 3.

Explanation of why each option is incorrect:

1. **Option 1 (2 zeros):** This option is incorrect because we have enough factors of 2 and 5 to form 3 pairs, not just 2. Therefore, there are more than 2 zeros.

3. **Option 3 (4 zeros):** This option is incorrect because, although we have 4 factors of 5, we only have 3 factors of 2. Hence, we cannot form 4 pairs of 2s and 5s, limiting us to 3 zeros.

4. **Option 4 (5 zeros):** This option is incorrect because we do not have enough factors of 2 to form 5 pairs with the factors of 5. We are limited by the 3 factors of 2, resulting in only 3 zeros.

In conclusion, the correct answer is Option 2, which states that there are 3 consecutive zeros at the end of the number.