Let's find the Least Common Multiple (LCM) for each set of numbers:

1. LCM of 7 and 21:

   • The prime factorization of 7 is $7$.
   • The prime factorization of 21 is $3 \times 7$.
   • The LCM is the highest power of each prime factor: $3 \times 7 = 21$.

   LCM(7, 21) = 21

2. LCM of 4, 8, and 12:

   • The prime factorization of 4 is $2^2$.
   • The prime factorization of 8 is $2^3$.
   • The prime factorization of 12 is $2^2 \times 3$.
   • The LCM is $2^3 \times 3 = 24$.

   LCM(4, 8, 12) = 24

3. LCM of 10, 15, and 45:

   • The prime factorization of 10 is $2 \times 5$.
   • The prime factorization of 15 is $3 \times 5$.
   • The prime factorization of 45 is $3^2 \times 5$.
   • The LCM is $2 \times 3^2 \times 5 = 90$.

   LCM(10, 15, 45) = 90

4. LCM of 12, 18, and 36:

   • The prime factorization of 12 is $2^2 \times 3$.
   • The prime factorization of 18 is $2 \times 3^2$.
   • The prime factorization of 36 is $2^2 \times 3^2$.
   • The LCM is $2^2 \times 3^2 = 36$.

   LCM(12, 18, 36) = 36

5. LCM of 9, 12, and 18:

   • The prime factorization of 9 is $3^2$.
   • The prime factorization of 12 is $2^2 \times 3$.
   • The prime factorization of 18 is $2 \times 3^2$.
   • The LCM is $2^2 \times 3^2 = 36$.

   LCM(9, 12, 18) = 36

Would you like further details on how these were calculated or do you have any questions?

Here are some related questions to expand on the topic:

1. How do you find the GCD (Greatest Common Divisor) of a set of numbers?
2. What is the relationship between the LCM and GCD of two numbers?
3. How can the LCM be used in solving fraction addition problems?
4. What is the significance of prime factorization in finding the LCM?
5. Can you find the LCM of algebraic expressions?

Tip: When finding the LCM, always remember to use the highest power of each prime factor present in the numbers.