To find the number of trailing zeros in a product, we need to count how many times 10 divides into the number. Each 10 is a product of 2 and 5, so we need to determine how many factors of 5 and 2 appear in the factorization of the entire expression.

Let’s break down the expression:

$(23 \times 34 \times 56 \times 35 \times 57 \times 79 \times 810) + (45 \times 56 \times 67 \times 78) - (102 \times 153 \times 204)$

Step 1: Prime Factorization

To find trailing zeros, we focus on the prime factorizations of each term and count the factors of 2 and 5.

Term 1: $23 \times 34 \times 56 \times 35 \times 57 \times 79 \times 810$

Prime factorizations:

• $23$ is prime.
• $34 = 2 \times 17$
• $56 = 2^3 \times 7$
• $35 = 5 \times 7$
• $57 = 3 \times 19$
• $79$ is prime.
• $810 = 2 \times 3^4 \times 5$

Factors of 2: $2 \times 2^3 \times 2 = 2^5$

Factors of 5: $5 \times 5 = 5^2$

Term 2: $45 \times 56 \times 67 \times 78$

Prime factorizations:

• $45 = 3^2 \times 5$
• $56 = 2^3 \times 7$
• $67$ is prime.
• $78 = 2 \times 3 \times 13$

Factors of 2: $2^3 \times 2 = 2^4$

Factors of 5: $5$

Term 3: $102 \times 153 \times 204$

Prime factorizations:

• $102 = 2 \times 3 \times 17$
• $153 = 3^2 \times 17$
• $204 = 2^2 \times 3 \times 17$

Factors of 2: $2 \times 2^2 = 2^3$

No factors of 5.

Step 2: Combine the Factors

Now, we combine the factors of 2 and 5 from each term.

• Total factors of 2:
  • From Term 1: $2^5$
  • From Term 2: $2^4$
  • From Term 3: $2^3$

Total factors of 2: $2^{5+4+3} = 2^{12}$

• Total factors of 5:
  • From Term 1: $5^2$
  • From Term 2: $5$
  • From Term 3: $0$

Total factors of 5: $5^{2+1} = 5^3$

Step 3: Determine the Number of Trailing Zeros

The number of trailing zeros is determined by the number of pairs of factors of 2 and 5. Since the number of factors of 5 is smaller, the number of trailing zeros is equal to the number of factors of 5.

Thus, the product has 3 trailing zeros.

---

Would you like further details or have any questions about this solution?

Here are 5 related questions for practice:

1. How do you determine the number of trailing zeros in a factorial?
2. What is the prime factorization of a large number like 810?
3. How do trailing zeros relate to divisibility by powers of 10?
4. What role does the highest power of 5 play in determining trailing zeros in large numbers?
5. How can we extend this technique to find trailing zeros in decimal fractions?

Tip: When calculating trailing zeros in large products, always focus on counting factors of 5 and 2 in the prime factorization of the numbers involved.