2.4 - Common Multiples? Using GCF and LCM to Solve Problems EQ: How do you recognize how to use common factors and common multiples to solve problems?

You have learned how to determine common factors, common multiples, greatest common factors, and least common multiples. Sometimes it is necessary to decide whether you need to use common factors or common multiples. How can you determine when to use common factors or common multiples to solve problems? Can you think of any real-world examples of when you would use common factors or common multiples?

Remember… Common factors help you think about how to divide, or share things equally. Common multiples help you think about how things with different cycles can occur at the same time.

Question Sabrina is watching one of her favorite television shows. She notices that there is a commercial every nine minutes. Her brother, Michael, is watching one of his favorite shows on another television station at the same time. He notices that there is a commercial every 12 minutes. Sabrina and Michael wonder if there will ever be a commercial on both channels at the same time. If Sabrina’s show begins at 8:00 pm and lasts for two hours, list all the times commercials will air during the show. Commercials will air at the following times: 8:09, 8:18, 8:27, 8:36, 8:45, 8:54, 9:03, 9:12, 9:21, 9:30, 9:39, 9:48, and 9:57.

Question Sabrina is watching one of her favorite television shows. She notices that there is a commercial every nine minutes. Her brother, Michael, is watching one of his favorite shows on another television station at the same time. He notices that there is a commercial every 12 minutes. Sabrina and Michael wonder if there will ever be a commercial on both channels at the same time. Michael’s show starts at the same time and also lasts for two hours. List all the times commercials will air during the show. Commercials will air at the following times: 8:12, 8:24, 8:36, 8:48, 9:00, 9:12, 9:24, 9:36, and 9:48.

Question Sabrina is watching one of her favorite television shows. She notices that there is a commercial every nine minutes. Her brother, Michael, is watching one of his favorite shows on another television station at the same time. He notices that there is a commercial every 12 minutes. Sabrina and Michael wonder if there will ever be a commercial on both channels at the same time. Will Sabrina’s and Michael’s shows have commercials air at the same time? If yes, how many times will Sabrina’s and Michael’s television shows air commercials at the same time, and at what times will this occur? Yes. Both Sabrina’s and Michael’s television shows will air commercials at the same time three different times: at 8:36, 9:12, and 9:48.

Question Sabrina is watching one of her favorite television shows. She notices that there is a commercial every nine minutes. Her brother, Michael, is watching one of his favorite shows on another television station at the same time. He notices that there is a commercial every 12 minutes. Sabrina and Michael wonder if there will ever be a commercial on both channels at the same time. How many minutes will it take for commercials to air on both stations at the same time? Why? It will take 36 minutes for commercials to air on both stations because 36 is the least common multiple of 9 and 12

Your Turn Ramon and Justine are watching different broadcasts of the Rose Parade on television. The broadcast Ramon is watching airs commercials every 17 minutes. The broadcast Justine is watching airs commercials every 14 minutes. Both broadcasts started at 7:00 am and are scheduled to end at 9:00 am. When will commercials air on both broadcasts at the same time? Explain your reasoning. The broadcasts will never have commercials air at the same time. The least common multiple for 17 and 14 is 238. In terms of the problem situation, this means that commercials will air at the same time in 238 minutes; however, the broadcasts are each only two hours, or 120 minutes long. Therefore, the broadcasts will never air commercials at the same time.

Your Turn Two bikers ride on the same circular path. The first rider completes a lap in 12 minutes. The second rider completes a lap in 18 minutes. Both riders start at the starting line at the same time and go in the same direction. If the riders maintain their speed, after how many minutes will they meet again at the starting line? Explain your reasoning. The least common multiple for 12 and 18 is 36; therefore, they will meet again at the starting line in 36 minutes.

Question Determine the greatest common factor by listing the factors. Dulcina has 45 pencils and 36 markers. She wants to make back-to-school packs that contain the same amount of pencils and the same amount of markers to give to her friends. Dulcina does not want any leftover pencils or markers. She knows that she needs to determine the greatest common factor. Explain to Dulcina how to determine the greatest common factor in three different ways. Determine the greatest common factor by listing the factors. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors are 1, 3, and 9. I can tell Dulcina that when she looks at the list of factors, 9 is the greatest common factor.

Question Determine the greatest common factor by using a Venn diagram. Dulcina has 45 pencils and 36 markers. She wants to make back-to-school packs that contain the same amount of pencils and the same amount of markers to give to her friends. Dulcina does not want any leftover pencils or markers. She knows that she needs to determine the greatest common factor. Explain to Dulcina how to determine the greatest common factor in three different ways. Determine the greatest common factor by using a Venn diagram.

Question Determine the greatest common factor by using a Venn diagram. Dulcina has 45 pencils and 36 markers. She wants to make back-to-school packs that contain the same amount of pencils and the same amount of markers to give to her friends. Dulcina does not want any leftover pencils or markers. She knows that she needs to determine the greatest common factor. Explain to Dulcina how to determine the greatest common factor in three different ways. Determine the greatest common factor by using a Venn diagram.

Question Dulcina has 45 pencils and 36 markers. She wants to make back-to-school packs that contain the same amount of pencils and the same amount of markers to give to her friends. Dulcina does not want any leftover pencils or markers. She knows that she needs to determine the greatest common factor. Explain to Dulcina how to determine the greatest common factor in three different ways. Demonstrate the greatest common factor using prime factorization. If you need help, create a factor tree. The prime factorization of 36 is 2 3 2 3 3 3 3, or 22 3 32. The prime factorization of 45 is 3 3 3 3 5, or 32 3 5. To determine the greatest common factor from the prime factorization, the factor must appear in both prime factorizations. So, I can tell Dulcina that 32 is the greatest common factor of 36 and 45, and that 32 is equal to 9.

Question Which method do you prefer? Why? Dulcina has 45 pencils and 36 markers. She wants to make back-to-school packs that contain the same amount of pencils and the same amount of markers to give to her friends. Dulcina does not want any leftover pencils or markers. She knows that she needs to determine the greatest common factor. Explain to Dulcina how to determine the greatest common factor in three different ways. Which method do you prefer? Why?