Least Common Multiples of Numbers to 100 from Prime Factors with ... - Free Printable

You’ve uploaded a worksheet titled “Least Common Multiple (A)” that asks students to find the LCM (Least Common Multiple) of pairs of numbers using their prime factorizations.

Since you asked me to “solve the problem and explain the solution,” I’ll walk through the method used in this worksheet, then verify and explain each problem shown — even though all answers are already filled in. This will help you understand *how* to solve any similar LCM problem using prime factors.

---

## ✔ What is LCM?
The Least Common Multiple of two or more numbers is the smallest number that is a multiple of each of them.

---

## ✔ How to Find LCM Using Prime Factorization

Step-by-step method:

1. Factor each number into its prime factors (as given in the worksheet).
2. List all the prime bases that appear in *any* of the factorizations.
3. For each prime base, take the highest exponent that appears in any of the factorizations.
4. Multiply those together to get the LCM.

> 💡 Think of it like building the smallest “common” number that contains *all* the prime factors needed by *each* original number.

---

## 🔍 Let’s Go Through Each Problem

---

Problem 1:

- 90 = 2 × 3² × 5
- 39 = 3 × 13

→ Primes involved: 2, 3, 5, 13
→ Highest exponents:
  2¹, 3², 5¹, 13¹
→ LCM = 2 × 3² × 5 × 13 = 1170 ✔

---

Problem 2:

- 70 = 2 × 5 × 7
- 60 = 2² × 3 × 5

→ Primes: 2, 3, 5, 7
→ Highest exponents: 2², 3¹, 5¹, 7¹
→ LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420 ✔

---

Problem 3:

- 16 = 2⁴
- 28 = 2² × 7

→ Primes: 2, 7
→ Highest exponents: 2⁴, 7¹
→ LCM = 2⁴ × 7 = 16 × 7 = 112 ✔

---

Problem 4:

- 76 = 2² × 19
- 86 = 2 × 43

→ Primes: 2, 19, 43
→ Highest exponents: 2², 19¹, 43¹
→ LCM = 2² × 19 × 43 = 4 × 19 × 43
→ 4 × 19 = 76 → 76 × 43 = 3268 ✔

*(Note: 76 × 40 = 3040; 76 × 3 = 228 → 3040 + 228 = 3268)*

---

Problem 5:

- 68 = 2² × 17
- 58 = 2 × 29

→ Primes: 2, 17, 29
→ Highest exponents: 2², 17¹, 29¹
→ LCM = 2² × 17 × 29 = 4 × 17 × 29
→ 4 × 17 = 68 → 68 × 29
→ 68 × 30 = 2040 → minus 68 → 1972 ✔

---

Problem 6:

- 66 = 2 × 3 × 11
- 10 = 2 × 5

→ Primes: 2, 3, 5, 11
→ Highest exponents: 2¹, 3¹, 5¹, 11¹
→ LCM = 2 × 3 × 5 × 11 = 30 × 11 = 330 ✔

---

Problem 7:

- 56 = 2³ × 7
- 46 = 2 × 23

→ Primes: 2, 7, 23
→ Highest exponents: 2³, 7¹, 23¹
→ LCM = 2³ × 7 × 23 = 8 × 7 × 23
→ 8 × 7 = 56 → 56 × 23
→ 50×23=1150, 6×23=138 → 1150+138 = 1288 ✔

---

Problem 8:

- 39 = 3 × 13
- 57 = 3 × 19

→ Primes: 3, 13, 19
→ Highest exponents: 3¹, 13¹, 19¹
→ LCM = 3 × 13 × 19
→ 3 × 13 = 39 → 39 × 19
→ 40×19=760 minus 19 = 741 ✔

---

Problem 9:

- 20 = 2² × 5
- 48 = 2⁴ × 3

→ Primes: 2, 3, 5
→ Highest exponents: 2⁴, 3¹, 5¹
→ LCM = 2⁴ × 3 × 5 = 16 × 3 × 5 = 48 × 5 = 240 ✔

---

Problem 10:

- 74 = 2 × 37
- 4 = 2²

→ Primes: 2, 37
→ Highest exponents: 2², 37¹
→ LCM = 2² × 37 = 4 × 37 = 148 ✔

---

## 🎯 Summary: Key Takeaway

To find LCM using prime factorization:

> Take every prime that appears in any factorization, raise it to the highest power it has in any of the numbers, and multiply them all together.

This method is reliable, systematic, and works for any set of numbers — not just two!

---

✔ All answers on the worksheet are correct. The red text shows the correct LCM calculation using the prime factor method.

Let me know if you’d like to practice with new numbers or need a printable version!