To find how many factors $10800$ has, we first perform its prime factorization, then use the formula for determining the number of factors.

Step 1: Prime Factorization of 10800

We can start by factoring $10800$:

$10800 \div 2 = 5400$ $5400 \div 2 = 2700$ $2700 \div 2 = 1350$ $1350 \div 2 = 675 \quad (\text{675 is not divisible by } 2)$ $675 \div 3 = 225$ $225 \div 3 = 75$ $75 \div 3 = 25$ $25 \div 5 = 5$ $5 \div 5 = 1$

Thus, $10800 = 2^4 \times 3^3 \times 5^2$.

Step 2: Finding the Number of Factors

The formula for determining the number of factors of a number from its prime factorization is:

$\text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \cdots$ where $e_1, e_2, e_3, \dots$ are the exponents of the prime factors.

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

$\text{Number of factors} = (4+1)(3+1)(2+1) = 5 \times 4 \times 3 = 60$

Conclusion:

The number of factors of $10800$ is 60.

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Would you like further details on how factorization works or how this can be applied to different contexts?

Here are some related questions to explore:

1. What are all the factors of $10800$?
2. How do prime factorizations help in finding the greatest common divisor (GCD)?
3. How does the prime factorization method differ from using divisibility rules?
4. Can factorization help solve problems involving least common multiple (LCM)?
5. What are the properties of numbers with exactly 60 factors?

Tip: Remember that each distinct factor of a number is a combination of powers of its prime factors!