3 Use prime factors to determine whether 18 is a factor of 252.

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UpStudy AI · Accepted Answer

To determine whether 18 is a factor of 252 using prime factors, we will follow these steps:

1. **Find the prime factorization of both numbers.**
2. **Check if the prime factors of 18 are present in the prime factorization of 252.**

### Step 1: Prime Factorization

**Prime factorization of 18:**
- 18 can be divided by 2 (the smallest prime number):
  $$
  18 \div 2 = 9
  $$
- Next, we factor 9, which can be divided by 3:
  $$
  9 \div 3 = 3
  $$
- Finally, we factor 3, which is a prime number.

Thus, the prime factorization of 18 is:
$$
18 = 2^1 \times 3^2
$$

**Prime factorization of 252:**
- 252 can be divided by 2:
  $$
  252 \div 2 = 126
  $$
- 126 can also be divided by 2:
  $$
  126 \div 2 = 63
  $$
- Next, we factor 63, which can be divided by 3:
  $$
  63 \div 3 = 21
  $$
- 21 can also be divided by 3:
  $$
  21 \div 3 = 7
  $$
- Finally, 7 is a prime number.

Thus, the prime factorization of 252 is:
$$
252 = 2^2 \times 3^2 \times 7^1
$$

### Step 2: Check for Factors

Now we compare the prime factorizations:
- For 18: $$2^1 \times 3^2$$
- For 252: $$2^2 \times 3^2 \times 7^1$$

To determine if 18 is a factor of 252, we check if the prime factors of 18 are present in the prime factorization of 252:
- The factor $$2^1$$ in 18 is present in $$2^2$$ in 252 (sufficient).
- The factor $$3^2$$ in 18 is present in $$3^2$$ in 252 (sufficient).

Since all prime factors of 18 are present in the prime factorization of 252, we conclude that:

**18 is a factor of 252.**
18 is a factor of 252.

3 Use prime factors to determine whether 18 is a factor of 252.

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