Worksheet for finding the greatest common factor (G.C.F.) of monomials, featuring eight problems with algebraic expressions.

To find the Greatest Common Factor (G.C.F.) of monomials, we need to identify the common factors in the coefficients and the lowest powers of the variables present in all the given monomials. Let's solve each problem step by step.

---

Problem 1: \( a^2bc, b^2c, c^2a \)

- Step 1: Identify the common variables.
- Variables in \( a^2bc \): \( a, b, c \)
- Variables in \( b^2c \): \( b, c \)
- Variables in \( c^2a \): \( a, c \)
- Common variables: \( c \)

- Step 2: Determine the lowest power of the common variable.
- Power of \( c \) in \( a^2bc \): 1
- Power of \( c \) in \( b^2c \): 1
- Power of \( c \) in \( c^2a \): 2
- Lowest power of \( c \): 1

- Step 3: Combine the results.
- G.C.F.: \( c \)

Answer: \( c \)

---

Problem 2: \( p^2q^2s^2, q^2r^2s^2, pqr \)

- Step 1: Identify the common variables.
- Variables in \( p^2q^2s^2 \): \( p, q, s \)
- Variables in \( q^2r^2s^2 \): \( q, r, s \)
- Variables in \( pqr \): \( p, q, r \)
- Common variables: \( q \)

- Step 2: Determine the lowest power of the common variable.
- Power of \( q \) in \( p^2q^2s^2 \): 2
- Power of \( q \) in \( q^2r^2s^2 \): 2
- Power of \( q \) in \( pqr \): 1
- Lowest power of \( q \): 1

- Step 3: Combine the results.
- G.C.F.: \( q \)

Answer: \( q \)

---

Problem 3: \( Qrt, rto, pqr \)

- Step 1: Identify the common variables.
- Variables in \( Qrt \): \( Q, r, t \)
- Variables in \( rto \): \( r, t, o \)
- Variables in \( pqr \): \( p, q, r \)
- Common variables: None

- Step 2: Since there are no common variables, the G.C.F. is 1.

Answer: \( 1 \)

---

Problem 4: \( wxy, x^2y^2, xwy^3 \)

- Step 1: Identify the common variables.
- Variables in \( wxy \): \( w, x, y \)
- Variables in \( x^2y^2 \): \( x, y \)
- Variables in \( xwy^3 \): \( w, x, y \)
- Common variables: \( x, y \)

- Step 2: Determine the lowest power of the common variables.
- Power of \( x \) in \( wxy \): 1
- Power of \( x \) in \( x^2y^2 \): 2
- Power of \( x \) in \( xwy^3 \): 1
- Lowest power of \( x \): 1
- Power of \( y \) in \( wxy \): 1
- Power of \( y \) in \( x^2y^2 \): 2
- Power of \( y \) in \( xwy^3 \): 3
- Lowest power of \( y \): 1

- Step 3: Combine the results.
- G.C.F.: \( xy \)

Answer: \( xy \)

---

Problem 5: \( mno, m^2n^2, n^2o^2m \)

- Step 1: Identify the common variables.
- Variables in \( mno \): \( m, n, o \)
- Variables in \( m^2n^2 \): \( m, n \)
- Variables in \( n^2o^2m \): \( m, n, o \)
- Common variables: \( m, n \)

- Step 2: Determine the lowest power of the common variables.
- Power of \( m \) in \( mno \): 1
- Power of \( m \) in \( m^2n^2 \): 2
- Power of \( m \) in \( n^2o^2m \): 1
- Lowest power of \( m \): 1
- Power of \( n \) in \( mno \): 1
- Power of \( n \) in \( m^2n^2 \): 2
- Power of \( n \) in \( n^2o^2m \): 2
- Lowest power of \( n \): 1

- Step 3: Combine the results.
- G.C.F.: \( mn \)

Answer: \( mn \)

---

Problem 6: \( a^2b^2m^2, m^2n^2, b^2n^2 \)

- Step 1: Identify the common variables.
- Variables in \( a^2b^2m^2 \): \( a, b, m \)
- Variables in \( m^2n^2 \): \( m, n \)
- Variables in \( b^2n^2 \): \( b, n \)
- Common variables: None

- Step 2: Since there are no common variables, the G.C.F. is 1.

Answer: \( 1 \)

---

Problem 7: \( c^3b^3a^3, abc, a^2b^2c^2 \)

- Step 1: Identify the common variables.
- Variables in \( c^3b^3a^3 \): \( a, b, c \)
- Variables in \( abc \): \( a, b, c \)
- Variables in \( a^2b^2c^2 \): \( a, b, c \)
- Common variables: \( a, b, c \)

- Step 2: Determine the lowest power of the common variables.
- Power of \( a \) in \( c^3b^3a^3 \): 3
- Power of \( a \) in \( abc \): 1
- Power of \( a \) in \( a^2b^2c^2 \): 2
- Lowest power of \( a \): 1
- Power of \( b \) in \( c^3b^3a^3 \): 3
- Power of \( b \) in \( abc \): 1
- Power of \( b \) in \( a^2b^2c^2 \): 2
- Lowest power of \( b \): 1
- Power of \( c \) in \( c^3b^3a^3 \): 3
- Power of \( c \) in \( abc \): 1
- Power of \( c \) in \( a^2b^2c^2 \): 2
- Lowest power of \( c \): 1

- Step 3: Combine the results.
- G.C.F.: \( abc \)

Answer: \( abc \)

---

Problem 8: \( x^3y^3z^3, xyz, x^2y^2z^2 \)

- Step 1: Identify the common variables.
- Variables in \( x^3y^3z^3 \): \( x, y, z \)
- Variables in \( xyz \): \( x, y, z \)
- Variables in \( x^2y^2z^2 \): \( x, y, z \)
- Common variables: \( x, y, z \)

- Step 2: Determine the lowest power of the common variables.
- Power of \( x \) in \( x^3y^3z^3 \): 3
- Power of \( x \) in \( xyz \): 1
- Power of \( x \) in \( x^2y^2z^2 \): 2
- Lowest power of \( x \): 1
- Power of \( y \) in \( x^3y^3z^3 \): 3
- Power of \( y \) in \( xyz \): 1
- Power of \( y \) in \( x^2y^2z^2 \): 2
- Lowest power of \( y \): 1
- Power of \( z \) in \( x^3y^3z^3 \): 3
- Power of \( z \) in \( xyz \): 1
- Power of \( z \) in \( x^2y^2z^2 \): 2
- Lowest power of \( z \): 1

- Step 3: Combine the results.
- G.C.F.: \( xyz \)

Answer: \( xyz \)

---

Final Answers:

1. \( c \)
2. \( q \)
3. \( 1 \)
4. \( xy \)
5. \( mn \)
6. \( 1 \)
7. \( abc \)
8. \( xyz \)

\[
\boxed{c, q, 1, xy, mn, 1, abc, xyz}
\]