Finding HCF and LCM Using Prime Factorization worksheet | Live ... - Free Printable

Let’s solve each part step by step.

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Part A: 18 and 27

Step 1: Prime factorization using factor trees.

For 18:
- Start with smallest prime that divides 18 → 2
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
→ So, 18 = 2 × 3 × 3

For 27:
- Smallest prime that divides 27 → 3
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
→ So, 27 = 3 × 3 × 3

Step 2: Find HCF (Highest Common Factor)
- Look for common prime factors in both.
- 18: 2 × 3 × 3
- 27: 3 × 3 × 3
→ Common: 3 × 3 = 9

Step 3: Find LCM (Least Common Multiple)
- Take all primes from both, but don’t repeat more than needed.
- From 18: 2, 3, 3
- From 27: 3, 3, 3 → we already have two 3s from 18, so add one more 3
→ LCM = 2 × 3 × 3 × 3 = 54

✔ Check:
HCF(18,27) = 9 → correct
LCM(18,27) = 54 → because 54 ÷ 18 = 3, 54 ÷ 27 = 2 → correct

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Part B: 20 and 30

Step 1: Prime factorization

For 20:
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 ÷ 5 = 1
→ 20 = 2 × 2 × 5

For 30:
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
→ 30 = 2 × 3 × 5

Step 2: HCF
- Common primes: 2 and 5
→ HCF = 2 × 5 = 10

Step 3: LCM
- All primes: from 20 → 2,2,5; from 30 → 2,3,5
- Take highest count of each: 2², 3¹, 5¹
→ LCM = 2 × 2 × 3 × 5 = 60

✔ Check:
HCF(20,30)=10 → correct
LCM(20,30)=60 → 60÷20=3, 60÷30=2 → correct

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Part C: 24 and 36 (using ladder method)

Ladder method: divide both numbers by same prime until you can’t anymore.

Start with 24 and 36:

Divide by 2:
24 ÷ 2 = 12
36 ÷ 2 = 18

Divide by 2 again:
12 ÷ 2 = 6
18 ÷ 2 = 9

Now 6 and 9 → divide by 3:
6 ÷ 3 = 2
9 ÷ 3 = 3

Now 2 and 3 → no common prime → stop.

So, prime factors:

24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3

Step 2: HCF
- Common primes: 2, 2, 3 → 2×2×3 = 12

Step 3: LCM
- All primes: from 24 → 2,2,2,3; from 36 → 2,2,3,3
- Highest counts: 2³, 3² → 8 × 9 = 72

✔ Check:
HCF(24,36)=12 → correct
LCM(24,36)=72 → 72÷24=3, 72÷36=2 → correct

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Part D: 50 and 75 (ladder method)

Start with 50 and 75.

Divide by 5:
50 ÷ 5 = 10
75 ÷ 5 = 15

Divide by 5 again:
10 ÷ 5 = 2
15 ÷ 5 = 3

Now 2 and 3 → no common prime → stop.

Prime factors:

50 = 5 × 5 × 2
75 = 5 × 5 × 3

Step 2: HCF
- Common: 5 × 5 = 25

Step 3: LCM
- All primes: 5,5,2 from 50; 5,5,3 from 75 → take 5², 2, 3
→ LCM = 5 × 5 × 2 × 3 = 150

✔ Check:
HCF(50,75)=25 → correct
LCM(50,75)=150 → 150÷50=3, 150÷75=2 → correct

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Final Answer:

A.
18: 2 × 3 × 3
27: 3 × 3 × 3
HCF: 9
LCM: 54

B.
20: 2 × 2 × 5
30: 2 × 3 × 5
HCF: 10
LCM: 60

C.
24: 2 × 2 × 2 × 3
36: 2 × 2 × 3 × 3
HCF: 12
LCM: 72

D.
50: 2 × 5 × 5
75: 3 × 5 × 5
HCF: 25
LCM: 150