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Let's solve the H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) for each set of numbers using prime factorization.

### 1. For the numbers **20, 50, 80**:

**Step 1: Prime factorization**:
- \( 20 = 2^2 \times 5 \)
- \( 50 = 2 \times 5^2 \)
- \( 80 = 2^4 \times 5 \)

**Step 2: H.C.F. (Greatest Common Factor)**:
- Common factors are \( 2 \) and \( 5 \).
- The lowest powers are \( 2^1 \times 5^1 = 2 \times 5 = 10 \).

Thus, **H.C.F. = 10**.

**Step 3: L.C.M. (Lowest Common Multiple)**:
- The highest powers are \( 2^4 \times 5^2 = 16 \times 25 = 400 \).

Thus, **L.C.M. = 400**.

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### 2. For the numbers **48, 72, 108**:

**Step 1: Prime factorization**:
- \( 48 = 2^4 \times 3 \)
- \( 72 = 2^3 \times 3^2 \)
- \( 108 = 2^2 \times 3^3 \)

**Step 2: H.C.F.**:
- Common factors are \( 2 \) and \( 3 \).
- The lowest powers are \( 2^2 \times 3^1 = 4 \times 3 = 12 \).

Thus, **H.C.F. = 12**.

**Step 3: L.C.M.**:
- The highest powers are \( 2^4 \times 3^3 = 16 \times 27 = 432 \).

Thus, **L.C.M. = 432**.

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### 3. For the numbers **84, 108, 168**:

**Step 1: Prime factorization**:
- \( 84 = 2^2 \times 3 \times 7 \)
- \( 108 = 2^2 \times 3^3 \)
- \( 168 = 2^3 \times 3 \times 7 \)

**Step 2: H.C.F.**:
- Common factors are \( 2 \) and \( 3 \).
- The lowest powers are \( 2^2 \times 3^1 = 4 \times 3 = 12 \).

Thus, **H.C.F. = 12**.

**Step 3: L.C.M.**:
- The highest powers are \( 2^3 \times 3^3 \times 7 = 8 \times 27 \times 7 = 1512 \).

Thus, **L.C.M. = 1512**.

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### 4. For the numbers **72, 90, 180**:

**Step 1: Prime factorization**:
- \( 72 = 2^3 \times 3^2 \)
- \( 90 = 2 \times 3^2 \times 5 \)
- \( 180 = 2^2 \times 3^2 \times 5 \)

**Step 2: H.C.F.**:
- Common factors are \( 2 \) and \( 3^2 \).
- The lowest powers are \( 2^1 \times 3^2 = 2 \times 9 = 18 \).

Thus, **H.C.F. = 18**.

**Step 3: L.C.M.**:
- The highest powers are \( 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 \).

Thus, **L.C.M. = 360**.

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Let me know if you'd like any further details on the steps.

### 5 Related Questions:
1. How can you use the H.C.F. and L.C.M. in solving word problems?
2. How do you verify the L.C.M. using division by prime numbers method?
3. What is the relationship between H.C.F. and L.C.M. of two numbers?
4. How can you use H.C.F. to simplify fractions?
5. How can finding prime factors help in solving problems involving divisibility?

### Tip:
Remember that the H.C.F. helps in simplifying common factors, while the L.C.M. is used when working with multiples, especially when aligning periodic events.

Find the H.C.F. and L.C.M. of the following sets of numbers by using prime factorization: (1) 20, 50, 80 (2) 48, 72, 108 (3) 84, 108, 168 (4) 72, 90, 180

Learn how to calculate the HCF and LCM of sets of numbers like 20, 50, 80 and 48, 72, 108 using prime factorization. This math exercise covers essential concepts for students in Grades 6-8, including the prime factorization method.

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