Cups are sold 5 to a package and plates are sold 10 to a package. If you want to have the same number of each item for the party, what is the least number of packages of each you need to buy?
To determine the least number of packages of cups and plates you need to buy to have the same number of each item for the party, we need to find the least common multiple (LCM) of the numbers 5 and 10.
Step 1: Find the prime factorization of each number:
- 5 = 5
- 10 = 2 * 5
Step 2: Write down the prime factors with the highest exponent for each number:
- From the prime factorization of 5, the highest exponent is 1.
- From the prime factorization of 10, the highest exponent is 1 for the prime factor 2 and 1 for the prime factor 5.
Step 3: Multiply the prime factors with their highest exponents together:
2^1 * 5^1 = 10
The LCM of 5 and 10 is 10.
Therefore, you need to buy at least 1 package of cups (5 cups in a package) and 1 package of plates (10 plates in a package) to have the same number of each item for the party.
To find the least number of packages of cups and plates needed to have the same number of each item, we need to find the least common multiple (LCM) of the quantities in each package.
The least common multiple is the smallest multiple that two or more numbers have in common. In this case, we need to find the LCM of 5 and 10.
To find the LCM of two numbers, we can list the multiples of each number until we find a common multiple. Let's do that for 5 and 10:
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, ...
From the lists above, we can see that the least common multiple of 5 and 10 is 10. This means that to have the same number of cups and plates, you would need to buy at least 1 package of cups (5 cups) and 1 package of plates (10 plates).