What are the factors of 210?

Steps to Solve

1. Find the prime factorization of 210

To start, we can break 210 down into its prime factors. This involves dividing by the smallest prime number first.

First, divide by 2 (the smallest prime): $$ 210 \div 2 = 105 $$ Next, we continue with 105, trying the next smallest prime, which is 3: $$ 105 \div 3 = 35 $$ Now, we factor 35 by using the next smallest prime number, which is 5: $$ 35 \div 5 = 7 $$ Finally, 7 is a prime number itself.

Thus, the prime factorization of 210 is: $$ 210 = 2 \times 3 \times 5 \times 7 $$

2. Identify the factors from the prime factorization

Now that we have the prime factorization, we can find all the factors of 210. To do this, we multiply the prime factors in all possible combinations.

The factors are:

• Start from $1 = 2^0 \times 3^0 \times 5^0 \times 7^0$
• Use combinations of the primes:
• $2^0 \times 3^0 \times 5^0 \times 7^0 = 1$
• $2^1 \times 3^0 \times 5^0 \times 7^0 = 2$
• $2^0 \times 3^1 \times 5^0 \times 7^0 = 3$
• $2^0 \times 3^0 \times 5^1 \times 7^0 = 5$
• $2^0 \times 3^0 \times 5^0 \times 7^1 = 7$
• Continuing this process until all combinations are exhausted.

3. List all the factors

By going through all combinations of the prime factors (and their exponents), we find the complete list of factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.