You are making necklaces for your friends. You have 72 blue beads and 42 red beads. Each friend will riecive an identical necklace, and all your beads must be used
What is your question?
12 necklaces will be made
To make necklaces for your friends using all of the 72 blue beads and 42 red beads, you can follow these steps:
Step 1: Determine the number of friends you are making necklaces for.
- Let's assume you are making necklaces for "x" number of friends.
Step 2: Calculate the number of beads needed for each necklace.
- Since each necklace should be identical, divide the total number of beads by the number of friends.
- Blue beads needed for each necklace: 72 / x
- Red beads needed for each necklace: 42 / x
Step 3: Check if the division results in a whole number.
- The number of beads for each necklace should be a whole number, without any fractions or remainders.
- Ensure that both (72 / x) and (42 / x) are whole numbers.
Step 4: Find the smallest value of x that satisfies the condition from Step 3.
- You need to find the smallest value of x that divides both 72 and 42 evenly.
- The greatest common divisor (GCD) of 72 and 42 is 6.
- Therefore, the smallest value of x is 6, and each necklace will have 12 blue beads (72 / 6) and 7 red beads (42 / 6).
Step 5: Calculate the total number of beads needed.
- Multiply the number of beads needed for each necklace by the number of friends.
- Total blue beads needed: 12 * x
- Total red beads needed: 7 * x
Step 6: Check if you have enough beads.
- Verify that the total number of blue beads (12 * x) is equal to 72, and the total number of red beads (7 * x) is equal to 42.
- If both conditions are met, then you have enough beads to make necklaces for your friends.
Following these steps, you can make necklaces for your friends using all 72 blue beads and 42 red beads.
To determine how many necklaces you can make for your friends using all the beads, we need to find the greatest common divisor (GCD) of the number of blue beads (72) and red beads (42).
Step 1: Find the prime factorization of both numbers.
- The prime factorization of 72: 2^3 * 3^2
- The prime factorization of 42: 2 * 3 * 7
Step 2: Identify the common prime factors and their exponents.
- The common prime factors are 2 and 3.
- The exponents for the common prime factors are 2 and 1, respectively.
Step 3: Multiply the common prime factors with the lowest exponent.
- Multiplying 2 * 3 = 6
Therefore, the GCD of 72 and 42 is 6. This means you can make 6 identical necklaces for your friends using all the beads. Each necklace will have 12 blue beads (72 ÷ 6) and 7 red beads (42 ÷ 6).