How many factors does 42 have?

Steps to Solve

1. Prime Factorization of 42

Start by finding the prime factors of 42. We can do this by dividing 42 by the smallest prime numbers until we are left with 1.

$$ 42 \div 2 = 21 \quad \text{(2 is a prime factor)} $$

Next, we factor 21:

$$ 21 \div 3 = 7 \quad \text{(3 is a prime factor)} $$

Finally, since 7 is a prime number, we stop here. The complete prime factorization of 42 is:

$$ 42 = 2^1 \times 3^1 \times 7^1 $$

2. Using the Prime Factorization to Find the Number of Factors

To find the total number of factors (divisors) of a number from its prime factorization, we use the formula:

If a number $n$ is expressed as $p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}$, the total number of factors $T(n)$ can be found using:

$$ T(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) $$

For 42, we have:

• $p_1 = 2$, $e_1 = 1$
• $p_2 = 3$, $e_2 = 1$
• $p_3 = 7$, $e_3 = 1$

Plugging the exponents into the formula gives us:

$$ T(42) = (1 + 1)(1 + 1)(1 + 1) $$

3. Calculating the Total Number of Factors

Now we can compute the total number of factors:

$$ T(42) = 2 \times 2 \times 2 = 8 $$

Thus, the total number of factors of 42 is 8.