LCM Word Problems worksheet - Free Printable

To solve these problems, we will use the concept of the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers.

Problem 1:

Today, both the soccer team and the basketball team had games. The soccer team plays every 3 days and the basketball team plays every 5 days. When will both teams have games on the same day again?

- Step 1: Identify the frequencies of the games.
- Soccer team: Every 3 days
- Basketball team: Every 5 days

- Step 2: Find the LCM of 3 and 5.
- The multiples of 3 are: 3, 6, 9, 12, 15, 18, ...
- The multiples of 5 are: 5, 10, 15, 20, 25, ...
- The smallest common multiple is 15.

- Step 3: Interpret the result.
- Both teams will have games on the same day again in 15 days.

Answer: $\boxed{15}$

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Problem 2:

Denesha is thinking of a number that is divisible by both 9 and 7. What is the smallest number Denesha could be thinking of?

- Step 1: Identify the numbers.
- The number must be divisible by both 9 and 7.

- Step 2: Find the LCM of 9 and 7.
- The prime factorization of 9 is $3^2$.
- The prime factorization of 7 is $7$.
- The LCM is obtained by taking the highest power of each prime factor: $3^2 \times 7 = 9 \times 7 = 63$.

- Step 3: Interpret the result.
- The smallest number divisible by both 9 and 7 is 63.

Answer: $\boxed{63}$

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Problem 3:

The high school marching band rehearses with either 6 or 10 members in every line. What is the least number of people that can be in the marching band?

- Step 1: Identify the number of members per line.
- The band can have 6 or 10 members per line.

- Step 2: Find the LCM of 6 and 10.
- The prime factorization of 6 is $2 \times 3$.
- The prime factorization of 10 is $2 \times 5$.
- The LCM is obtained by taking the highest power of each prime factor: $2 \times 3 \times 5 = 30$.

- Step 3: Interpret the result.
- The least number of people in the marching band is 30.

Answer: $\boxed{30}$

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Problem 4:

Find the lowest number which is exactly divisible by 12 and 18.

- Step 1: Identify the numbers.
- The number must be divisible by both 12 and 18.

- Step 2: Find the LCM of 12 and 18.
- The prime factorization of 12 is $2^2 \times 3$.
- The prime factorization of 18 is $2 \times 3^2$.
- The LCM is obtained by taking the highest power of each prime factor: $2^2 \times 3^2 = 4 \times 9 = 36$.

- Step 3: Interpret the result.
- The lowest number divisible by both 12 and 18 is 36.

Answer: $\boxed{36}$

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Problem 5:

Find the lowest number which is exactly divisible by 14 and 16.

- Step 1: Identify the numbers.
- The number must be divisible by both 14 and 16.

- Step 2: Find the LCM of 14 and 16.
- The prime factorization of 14 is $2 \times 7$.
- The prime factorization of 16 is $2^4$.
- The LCM is obtained by taking the highest power of each prime factor: $2^4 \times 7 = 16 \times 7 = 112$.

- Step 3: Interpret the result.
- The lowest number divisible by both 14 and 16 is 112.

Answer: $\boxed{112}$

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Problem 6:

Steve can save 9 dollars every day while Maria can save 12 dollars every day. What is the least number of days it will take each person to save the same amount of money?

- Step 1: Identify the savings rates.
- Steve saves $9 per day.
- Maria saves $12 per day.

- Step 2: Find the LCM of 9 and 12.
- The prime factorization of 9 is $3^2$.
- The prime factorization of 12 is $2^2 \times 3$.
- The LCM is obtained by taking the highest power of each prime factor: $2^2 \times 3^2 = 4 \times 9 = 36$.

- Step 3: Determine the number of days.
- The LCM (36) represents the amount of money they will both save at the same time.
- For Steve: $36 \div 9 = 4$ days.
- For Maria: $36 \div 12 = 3$ days.
- The least number of days is the LCM of their individual times, which is 12 days (since they need to save the same amount).

Answer: $\boxed{12}$

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Final Answers:

1. $\boxed{15}$
2. $\boxed{63}$
3. $\boxed{30}$
4. $\boxed{36}$
5. $\boxed{112}$
6. $\boxed{12}$