Math  /  Numbers & Operations

Question92. A choral director needs to divede 180 men and 144 women into all-men and all-women singing groups so that each group has the same number of people. What is the largest number of people that can be placed in cach singing group?

Studdy Solution

STEP 1

1. The choral director wants to divide both men and women into groups where each group has the same number of people.
2. Each group must be all-men or all-women.
3. We need to find the largest number of people that can be placed in each group.

STEP 2

1. Identify the problem as finding the greatest common divisor (GCD).
2. Calculate the GCD of the two numbers: 180 and 144.
3. Interpret the GCD as the largest group size.

STEP 3

Recognize that the problem requires finding the largest number that divides both 180 and 144 evenly. This is the greatest common divisor (GCD).

STEP 4

List the prime factors of 180 and 144.
- For 180: - Divide by 2: 180÷2=90180 \div 2 = 90 - Divide by 2: 90÷2=4590 \div 2 = 45 - Divide by 3: 45÷3=1545 \div 3 = 15 - Divide by 3: 15÷3=515 \div 3 = 5 - Divide by 5: 5÷5=15 \div 5 = 1 - Prime factors of 180 are 22×32×52^2 \times 3^2 \times 5.
- For 144: - Divide by 2: 144÷2=72144 \div 2 = 72 - Divide by 2: 72÷2=3672 \div 2 = 36 - Divide by 2: 36÷2=1836 \div 2 = 18 - Divide by 2: 18÷2=918 \div 2 = 9 - Divide by 3: 9÷3=39 \div 3 = 3 - Divide by 3: 3÷3=13 \div 3 = 1 - Prime factors of 144 are 24×322^4 \times 3^2.

STEP 5

Identify the common prime factors and their lowest powers.
- Common prime factors are 22 and 33. - The lowest power of 22 is 222^2. - The lowest power of 33 is 323^2.

STEP 6

Calculate the GCD using the common prime factors.
GCD=22×32=4×9=36 \text{GCD} = 2^2 \times 3^2 = 4 \times 9 = 36

STEP 7

Interpret the GCD as the largest number of people that can be placed in each singing group.
The largest number of people that can be placed in each group is:
36\boxed{36}

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