If `y=30p`, and p is prime, what is the greatest common factor by y and 14p, in terms of p?

Answer

To find the greatest common factor (GCF) of \( y \) and \( 14p \) where \( y = 30p \) and \( p \) is a prime number, we can follow these steps:

### Step 1: Identify the expressions for \( y \) and \( 14p \)
We have:
- \( y = 30p \)
- \( 14p \)

### Step 2: Factor each expression into its prime factors
First, we need to factor \( 30p \):
- The prime factorization of \( 30 \) is \( 2 \times 3 \times 5 \).
- Therefore, the prime factorization of \( y = 30p \) is \( 2 \times 3 \times 5 \times p \).

Next, we factor \( 14p \):
- The prime factorization of \( 14 \) is \( 2 \times 7 \).
- Therefore, the prime factorization of \( 14p \) is \( 2 \times 7 \times p \).

### Step 3: Identify the common prime factors
Now we will find the common prime factors between \( 30p \) and \( 14p \):
- The prime factors of \( 30p \) are \( 2, 3, 5, \) and \( p \).
- The prime factors of \( 14p \) are \( 2, 7, \) and \( p \).

The common prime factors are:
- \( p \)
- \( 2 \)

### Step 4: Calculate the GCF
To find the GCF, we take the product of the common prime factors:
- The GCF is \( 2 \times p = 2p \).

### Conclusion
Thus, the greatest common factor of \( y \) and \( 14p \) is \( 2p \).