Questions and Answers
What is the prime factorization of 240?
2^3 × 3 × 5
What is the least common multiple (LCM) of 9 and 12?
36
What is the greatest common factor (GCF) of 48 and 60?
12
A bookshelf has 5 shelves, and each shelf can hold 8 books. How many books can the bookshelf hold in total?
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A rectangular garden has a length of 18 meters and a width of 24 meters. If a path that is 2 meters wide is built around the garden, what is the area of the path?
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In a factory, three machines produce identical parts at different rates. Machine A produces 12 parts every 30 minutes, Machine B produces 15 parts every 45 minutes, and Machine C produces 20 parts every 60 minutes. What is the least time it takes for all machines to produce the same number of parts?
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A company has three departments: A, B, and C. Department A has 12 employees, Department B has 15 employees, and Department C has 20 employees. If they want to distribute a certain number of bonuses equally among all employees, what is the least common multiple of the number of employees in each department?
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A group of friends want to go on a hike together. They want to find a common distance they can all hike without having to hike different distances. If Tom can hike 12 km, Rachel can hike 15 km, and Alex can hike 20 km, what is the least common multiple of their hiking distances?
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A bakery sells 240 loaves of bread per day. They pack 12 loaves of bread in a bag. How many bags of bread can they make in a day?
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A bakery sells a variety of bread in different quantities. They sell 12 whole wheat bread loaves in a pack, 15 white bread loaves in a pack, and 20 rye bread loaves in a pack. What is the least common multiple of the number of loaves in each pack?
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A rope is 15 meters long and needs to be cut into pieces of equal length. If the pieces need to be whole numbers of meters, what is the longest length of each piece that satisfies this condition?
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A school has two classes with 24 and 36 students respectively. The students need to be divided into groups of equal size. What is the largest number of students that can be in each group?
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Ashley has 12 bags of apples, and each bag contains 8 apples. If she wants to pack the apples into boxes of equal size, what is the smallest number of apples that can be in each box?
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A recipe requires a mixture of 2/3 cup of sugar and 3/4 cup of flour. What is the smallest common unit of measurement that can be used to measure both the sugar and the flour?
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A factory produces 24 units of a product every 3 hours, and 36 units of the same product every 4 hours. What is the smallest number of hours that the factory needs to produce a whole number of units of the product?
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Study Notes
LCM and GCF Applications
- LCM and GCF are used in real-world applications such as tiling a wall, finding the largest tile size without cutting, by finding the HCF of the wall and tile dimensions.
- LCM is used to determine the time it takes for two buses or trains to arrive together, by finding the LCM of their arrival times.
Multi-Step Problems
- Multi-step problems often involve finding the LCM or GCF of more than two numbers, such as in systems of equations, where you need to find the LCM of the denominators of the fractions involved.
Comparing Quantities
- When comparing quantities, the HCF is used to understand their relationship, such as finding the HCF of two numbers to find the largest number that divides both without leaving a remainder.
Finding Common Multiples
- Finding the LCM of two or more numbers is essential when dealing with multiples, such as finding the smallest unit in which two or more quantities can be measured.
- The HCF is used when dividing something up into equal parts.
Solving Systems
- In solving systems of equations, LCM or GCF is used to simplify the fractions involved, such as finding the LCM of the denominators to add or subtract fractions with different denominators.
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