To find the number of factors of 10,800, we first need to determine its prime factorization. Let's break down 10,800 into its prime factors:

Step 1: Prime Factorization of 10,800

We start by dividing 10,800 by small prime numbers:

$10800 \div 2 = 5400$ $5400 \div 2 = 2700$ $2700 \div 2 = 1350$ $1350 \div 2 = 675$ Now 675 is not divisible by 2, so we proceed with the next prime, 3:

$675 \div 3 = 225$ $225 \div 3 = 75$ $75 \div 3 = 25$ Now, 25 is not divisible by 3, so we move to the next prime, 5:

$25 \div 5 = 5$ $5 \div 5 = 1$

So, the prime factorization of 10,800 is: $10800 = 2^4 \times 3^3 \times 5^2$

Step 2: Finding the Number of Factors

To find the total number of factors, we use the formula for determining the number of divisors based on the prime factorization: $\text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \dots$ where $e_1, e_2, e_3, \dots$ are the exponents in the prime factorization.

For $10800 = 2^4 \times 3^3 \times 5^2$, the exponents are 4, 3, and 2.

The number of factors is: $(4+1)(3+1)(2+1) = 5 \times 4 \times 3 = 60$

Conclusion:

The number of factors of 10,800 is 60.

Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do we calculate the sum of the factors of 10,800?
2. What is the highest prime factor of 10,800?
3. How can you find the greatest common divisor (GCD) of 10,800 and another number?
4. How do we find the number of divisors for any composite number using its prime factorization?
5. What are the properties of numbers with exactly 60 divisors?

Tip: To find the number of factors of any number, use its prime factorization and apply the divisor formula.