1) Problem: 1530

Divide by 2.

$$\frac{1530}{2}=765$$

765 is not divisible by 2, so divide by 3.

$$\frac{765}{3}=255$$

Divide by 3 again.

$$\frac{255}{3}=85$$

85 is not divisible by 3, so divide by 5.

$$\frac{85}{5}=17$$

17 is already prime. Assemble a list of all the factors we divided by to find the prime factorization.

Answer: {2, 3, 3, 5, 17}

2) Problem: 31

31 is already a prime number, so it cannot be factored beyond itself.

Answer: 31

3) Problem: 1001

1001 is slightly tricky. Notice that it cannot be divided by 2, 3, or 5. So, move to the next largest prime, 7.

$$\frac{1001}{7}=143$$

143 cannot be divided by 7, but it can be divided by the next largest prime, 11.

$$\frac{143}{11}=13$$

17 is already prime. Assemble a list of all the factors we divided by to find the prime factorization.

Answer: {7, 11, 13}