1. **State the problem:** Find the greatest common factor (GCF) of $44by$ and $38$. 2. **Recall the definition:** The GCF of two numbers is the largest number that divides both without leaving a remainder. 3. **Factor each term into primes and variables:** - $44by = 44 \times b \times y = 2 \times 2 \times 11 \times b \times y$ - $38 = 2 \times 19$ 4. **Identify common prime factors:** Both have a factor of $2$. 5. **Variables:** $b$ and $y$ are only in the first term, so they are not common. 6. **Multiply common factors:** The only common factor is $2$. 7. **Conclusion:** The greatest common factor of $44by$ and $38$ is $2$.