Word Problems Multiples and factors worksheets - Math Worksheets ... - Free Printable

Let's solve each problem step by step.

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Problem 1:

At the gym, Hillary swims every 6 days, runs every 4 days, and cycles every 16 days. If she did all three activities today, in how many days will she do all three activities again on the same day?

#### Solution:
To find out when Hillary will do all three activities on the same day again, we need to determine the least common multiple (LCM) of the intervals at which she does each activity: 6 days (swimming), 4 days (running), and 16 days (cycling).

1. Prime factorization:
- \(6 = 2 \times 3\)
- \(4 = 2^2\)
- \(16 = 2^4\)

2. Find the LCM:
- The LCM is found by taking the highest power of each prime number that appears in the factorizations.
- For \(2\): The highest power is \(2^4\) (from 16).
- For \(3\): The highest power is \(3^1\) (from 6).
- Therefore, \(\text{LCM} = 2^4 \times 3 = 16 \times 3 = 48\).

#### Answer:
Hillary will do all three activities on the same day again in 48 days.

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Problem 2:

Oscar needs to ship 14 rock CDs, 12 classical CDs, and 8 pop CDs. He can pack only one type of CD in each box and he must pack the same number of CDs in each box. What is the greatest number of CDs Oscar can pack in each box?

#### Solution:
To determine the greatest number of CDs Oscar can pack in each box, we need to find the greatest common divisor (GCD) of the numbers of CDs: 14 (rock), 12 (classical), and 8 (pop).

1. Prime factorization:
- \(14 = 2 \times 7\)
- \(12 = 2^2 \times 3\)
- \(8 = 2^3\)

2. Find the GCD:
- The GCD is found by taking the lowest power of each prime number that appears in all factorizations.
- For \(2\): The lowest power is \(2^1\) (common in all three numbers).
- There are no other common prime factors.
- Therefore, \(\text{GCD} = 2\).

#### Answer:
The greatest number of CDs Oscar can pack in each box is 2.

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Problem 3:

I want to plant 45 sunflower plants, 81 corn plants, and 63 tomato plants in my garden. If I put the same number of plants in each row and each row has only one type of plant, what is the greatest number of plants I can put in one row?

#### Solution:
To find the greatest number of plants that can be put in each row, we need to determine the greatest common divisor (GCD) of the numbers of plants: 45 (sunflowers), 81 (corn), and 63 (tomatoes).

1. Prime factorization:
- \(45 = 3^2 \times 5\)
- \(81 = 3^4\)
- \(63 = 3^2 \times 7\)

2. Find the GCD:
- The GCD is found by taking the lowest power of each prime number that appears in all factorizations.
- For \(3\): The lowest power is \(3^2\) (common in all three numbers).
- There are no other common prime factors.
- Therefore, \(\text{GCD} = 3^2 = 9\).

#### Answer:
The greatest number of plants that can be put in one row is 9.

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Problem 4:

Cups are sold 6 to a package and plates are sold 8 to a package. If you want to have the same number of each item for a party, what is the least number of packages of each you need to buy?

#### Solution:
To have the same number of cups and plates, we need to find the least common multiple (LCM) of the package sizes: 6 (cups per package) and 8 (plates per package).

1. Prime factorization:
- \(6 = 2 \times 3\)
- \(8 = 2^3\)

2. Find the LCM:
- The LCM is found by taking the highest power of each prime number that appears in the factorizations.
- For \(2\): The highest power is \(2^3\) (from 8).
- For \(3\): The highest power is \(3^1\) (from 6).
- Therefore, \(\text{LCM} = 2^3 \times 3 = 8 \times 3 = 24\).

3. Determine the number of packages:
- To get 24 cups, you need \(\frac{24}{6} = 4\) packages of cups.
- To get 24 plates, you need \(\frac{24}{8} = 3\) packages of plates.

#### Answer:
The least number of packages needed is 4 packages of cups and 3 packages of plates.

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Problem 5:

A full moon occurs every 30 days. If the last full moon occurred on a Friday, how many days will pass before a full moon occurs again on a Friday?

#### Solution:
To determine when the next full moon will occur on a Friday, we need to consider both the cycle of full moons (every 30 days) and the cycle of days of the week (every 7 days). We are looking for the smallest number of days such that both cycles align again on a Friday.

1. Find the least common multiple (LCM) of 30 and 7:
- Prime factorization:
- \(30 = 2 \times 3 \times 5\)
- \(7 = 7\)
- The LCM is found by taking the highest power of each prime number that appears in the factorizations.
- For \(2\): The highest power is \(2^1\) (from 30).
- For \(3\): The highest power is \(3^1\) (from 30).
- For \(5\): The highest power is \(5^1\) (from 30).
- For \(7\): The highest power is \(7^1\) (from 7).
- Therefore, \(\text{LCM} = 2 \times 3 \times 5 \times 7 = 210\).

#### Answer:
The next full moon will occur on a Friday in 210 days.

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Final Answers:

1. \(\boxed{48}\)
2. \(\boxed{2}\)
3. \(\boxed{9}\)
4. \(\boxed{4 \text{ packages of cups and } 3 \text{ packages of plates}}\)
5. \(\boxed{210}\)