Let's go through the questions in the image you've uploaded:

Left side: Finding the lowest common multiple (LCM) of pairs of numbers

1. 8 and 6:

   • The prime factorizations are:
     • 8 = $2^3$
     • 6 = $2 \times 3$
   • The LCM is the product of the highest powers of all primes:
     • LCM(8, 6) = $2^3 \times 3 = 24$

2. 12 and 21:

   • The prime factorizations are:
     • 12 = $2^2 \times 3$
     • 21 = $3 \times 7$
   • LCM(12, 21) = $2^2 \times 3 \times 7 = 84$

3. 20 and 25:

   • The prime factorizations are:
     • 20 = $2^2 \times 5$
     • 25 = $5^2$
   • LCM(20, 25) = $2^2 \times 5^2 = 100$

4. 28 and 24:

   • The prime factorizations are:
     • 28 = $2^2 \times 7$
     • 24 = $2^3 \times 3$
   • LCM(28, 24) = $2^3 \times 3 \times 7 = 168$

5. 12 and 42:

   • The prime factorizations are:
     • 12 = $2^2 \times 3$
     • 42 = $2 \times 3 \times 7$
   • LCM(12, 42) = $2^2 \times 3 \times 7 = 84$

6. 50 and 20:

   • The prime factorizations are:
     • 50 = $2 \times 5^2$
     • 20 = $2^2 \times 5$
   • LCM(50, 20) = $2^2 \times 5^2 = 100$

Right side: City Hoppa bus problem (LCM of 25 and 45 minutes)

1. One bus has a route of 25 minutes, and the other has a route of 45 minutes. We need to find the next time they will be at the stop together, which is the LCM of 25 and 45:

   • The prime factorizations are:
     • 25 = $5^2$
     • 45 = $3^2 \times 5$
   • LCM(25, 45) = $3^2 \times 5^2 = 225$ minutes.

   225 minutes = 3 hours and 45 minutes.

   Since both buses leave at 6:00 a.m., they will both be at the stop again at 9:45 a.m..

Would you like more details or have any questions about the process?

5 Follow-up Questions:

1. What is the relationship between GCD (Greatest Common Divisor) and LCM?
2. Can you solve for the LCM of 15 and 40 using the prime factorization method?
3. How does knowing the prime factorization of a number help in finding its LCM?
4. If the first bus route changed to 30 minutes, how would that affect the next meeting time?
5. How would you calculate the LCM of more than two numbers, for example, 8, 12, and 15?

Tip:

Always break numbers into their prime factors to simplify finding LCM and GCD.