Worksheet on L.C.M. | Least Common Multiple Worksheets |LCM Worksheets - Free Printable

Let’s solve each problem step by step.

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Problem 1: Find the first 3 common multiples of the given using a number line.

(i) 3 and 4

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Common multiples (first 3): 12, 24, 36

✔ Check: 12 ÷ 3 = 4, 12 ÷ 4 = 3 → yes
24 ÷ 3 = 8, 24 ÷ 4 = 6 → yes
36 ÷ 3 = 12, 36 ÷ 4 = 9 → yes

(ii) 5 and 6

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, ...

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, ...

Common multiples (first 3): 30, 60, 90

✔ Check: 30 ÷ 5 = 6, 30 ÷ 6 = 5 → yes
60 ÷ 5 = 12, 60 ÷ 6 = 10 → yes
90 ÷ 5 = 18, 90 ÷ 6 = 15 → yes

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Problem 2: Find the LCM by prime factorization method.

(i) 12 and 20

Prime factors:

12 = 2 × 2 × 3 = 2² × 3
20 = 2 × 2 × 5 = 2² × 5

LCM = take highest power of each prime:
2² × 3 × 5 = 4 × 3 × 5 = 60

✔ Check: 60 ÷ 12 = 5, 60 ÷ 20 = 3 → yes

(ii) 36, 27 and 18

Prime factors:

36 = 2 × 2 × 3 × 3 = 2² × 3²
27 = 3 × 3 × 3 = 3³
18 = 2 × 3 × 3 = 2 × 3²

LCM = highest powers:
2² × 3³ = 4 × 27 = 108

✔ Check:
108 ÷ 36 = 3 → yes
108 ÷ 27 = 4 → yes
108 ÷ 18 = 6 → yes

(iii) 32 and 48

Prime factors:

32 = 2 × 2 × 2 × 2 × 2 = 2⁵
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

LCM = highest powers:
2⁵ × 3 = 32 × 3 = 96

✔ Check: 96 ÷ 32 = 3, 96 ÷ 48 = 2 → yes

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Problem 3: Find the L.C.M. by Long Division Method.

We divide all numbers together by common primes until no more common divisors exist.

(i) 70, 110, 150

Step-by-step division:

Divide by 2:
70 ÷ 2 = 35
110 ÷ 2 = 55
150 ÷ 2 = 75
→ Write down 2

Now: 35, 55, 75 — all divisible by 5? Yes.

Divide by 5:
35 ÷ 5 = 7
55 ÷ 5 = 11
75 ÷ 5 = 15
→ Write down 5

Now: 7, 11, 15 — no common divisor except 1.

But we can still divide any pair? No — for long division method, we stop when no prime divides ALL remaining numbers.

So multiply all divisors used: 2 × 5 = 10
Then multiply with remaining numbers: 7 × 11 × 15

Wait — actually, in long division method for LCM, you keep dividing as long as at least one number is divisible by the prime, but you only write the divisor if it divides *all* numbers? Actually, standard method: divide by primes that divide *at least two* numbers, and carry forward undivided ones.

Let me do it properly:

Start: 70, 110, 150

Divide by 2 (divides all):
→ 35, 55, 75 | record 2

Divide by 5 (divides all):
→ 7, 11, 15 | record 5

Now: 7, 11, 15 — no common prime divides all three.

But 15 is divisible by 3 and 5, others aren’t. So we stop here? Not quite — in some methods, you continue dividing even if not all are divisible, carrying forward unchanged numbers.

Actually, correct long division method for LCM:

You divide by smallest prime that divides *at least one* number, and bring down numbers not divisible.

Better to use this approach:

Write numbers: 70, 110, 150

Divide by 2 → 35, 55, 75 (since 2 divides all)

Divide by 5 → 7, 11, 15 (5 divides all)

Now: 7, 11, 15

No prime divides all three. But 15 is divisible by 3 and 5. Since 3 doesn't divide 7 or 11, we don’t divide by 3 yet? Actually, in LCM long division, we can divide by any prime that divides *any* of the numbers, and carry forward others.

Standard way:

After 2 and 5, we have 7, 11, 15

Now divide by 3? Only 15 is divisible → so divide 15 by 3 → get 5, leave 7 and 11 as is.

Record 3.

Now: 7, 11, 5

Divide by 5? Only 5 is divisible → divide 5 by 5 → get 1, leave 7 and 11.

Record 5.

Now: 7, 11, 1

Divide by 7? Only 7 → divide 7 by 7 → 1, leave 11.

Record 7.

Now: 1, 11, 1

Divide by 11 → 1, 1, 1

Record 11.

Now all 1s.

Multiply all recorded divisors: 2 × 5 × 3 × 5 × 7 × 11

Calculate step by step:

2 × 5 = 10
10 × 3 = 30
30 × 5 = 150
150 × 7 = 1050
1050 × 11 = 11550

That seems too big. Let me check with another method.

Alternative: Prime factorization of each:

70 = 2 × 5 × 7
110 = 2 × 5 × 11
150 = 2 × 3 × 5²

LCM = highest powers: 2¹, 3¹, 5², 7¹, 11¹
= 2 × 3 × 25 × 7 × 11

Calculate:

2 × 3 = 6
6 × 25 = 150
150 × 7 = 1050
1050 × 11 = 11550

Yes, same answer. So LCM is 11550.

But let me confirm with actual multiples? Too big, but math checks out.

Actually, wait — maybe I made a mistake in long division interpretation.

In many textbooks, for long division method for LCM of multiple numbers, you divide by primes that divide *at least two* numbers, and carry forward others.

Let me try again:

Numbers: 70, 110, 150

Divide by 2 (divides all) → 35, 55, 75 | record 2

Divide by 5 (divides all) → 7, 11, 15 | record 5

Now: 7, 11, 15

No prime divides all three. But 15 and ? 7 and 11 are prime. So we stop? Then LCM = 2 × 5 × 7 × 11 × 15? That would be wrong because 15 is not prime.

Actually, after getting 7, 11, 15, we should break down 15 further since it's composite.

So better to think: after dividing by common primes, multiply all divisors and the remaining numbers.

So after dividing by 2 and 5, we have 7, 11, 15.

But 15 = 3 × 5, so overall LCM = 2 × 5 × 7 × 11 × 3 × 5 = same as before: 2×3×5²×7×11 = 11550.

Yes.

So answer is 11550.

But let me see if there's a smaller common multiple? Unlikely, since 7, 11, and 25 (from 150) are involved.

Okay, moving on.

(ii) 25, 30, 150

Prime factorization might be faster, but let's do long division.

Numbers: 25, 30, 150

Divide by 2? 25 not divisible → skip? Or divide only those divisible? Standard method: divide by prime that divides at least one, carry forward others.

Better: find GCD first or use prime factors.

25 = 5²
30 = 2 × 3 × 5
150 = 2 × 3 × 5²

LCM = highest powers: 2¹, 3¹, 5² = 2 × 3 × 25 = 150

Check: 150 ÷ 25 = 6, 150 ÷ 30 = 5, 150 ÷ 150 = 1 → yes.

So LCM is 150.

Long division method:

Start: 25, 30, 150

Divide by 2? 25 not divisible → so divide 30 and 150 by 2 → get 15, 75; carry 25 down.

Record 2.

Now: 25, 15, 75

Divide by 3? 25 not divisible → divide 15 and 75 by 3 → get 5, 25; carry 25.

Record 3.

Now: 25, 5, 25

Divide by 5? All divisible → 5, 1, 5

Record 5.

Now: 5, 1, 5

Divide by 5? 5 and 5 divisible → 1, 1, 1; carry 1.

Record 5.

Now all 1s.

Multiply divisors: 2 × 3 × 5 × 5 = 150 → correct.

(iii) 36, 60, 120

Prime factors:

36 = 2² × 3²
60 = 2² × 3 × 5
120 = 2³ × 3 × 5

LCM = highest powers: 2³, 3², 5¹ = 8 × 9 × 5 = 72 × 5 = 360

Check: 360 ÷ 36 = 10, 360 ÷ 60 = 6, 360 ÷ 120 = 3 → yes.

Long division:

36, 60, 120

Divide by 2 → 18, 30, 60 | record 2

Divide by 2 → 9, 15, 30 | record 2

Divide by 2? 9 not divisible → divide 15 and 30 by 2? No, 15 odd. So stop dividing by 2.

Current: 9, 15, 30

Divide by 3 → 3, 5, 10 | record 3

Divide by 3? 3 divisible, 5 and 10 not → divide 3 by 3 → 1; carry 5,10

Record 3.

Now: 1, 5, 10

Divide by 5 → 1, 1, 2 | record 5

Now: 1, 1, 2

Divide by 2 → 1,1,1 | record 2

Multiply all: 2 × 2 × 3 × 3 × 5 × 2 = let's compute: 2×2=4, ×3=12, ×3=36, ×5=180, ×2=360 → correct.

So answers:

Problem 1(i): 12, 24, 36
Problem 1(ii): 30, 60, 90
Problem 2(i): 60
Problem 2(ii): 108
Problem 2(iii): 96
Problem 3(i): 11550
Problem 3(ii): 150
Problem 3(iii): 360

Final Answer:
1. (i) 12, 24, 36
(ii) 30, 60, 90
2. (i) 60
(ii) 108
(iii) 96
3. (i) 11550
(ii) 150
(iii) 360