Question
At a concert, the band has 49 men's T-shirts and 42 women's T-shirts. The band wants to set up tables to sell the shirts, with an equal number of men's and women's shirts available at each table and no shirts left over. What is the greatest number of tables the band can sell shirts from
Answers
GPT-4o mini
To determine the greatest number of tables the band can set up to sell the T-shirts, we need to find the greatest common divisor (GCD) of the number of men's T-shirts and women's T-shirts.
The numbers of T-shirts are:
- Men's T-shirts: 49
- Women's T-shirts: 42
First, we will find the prime factorization of both numbers.
**Prime factorization of 49**:
\[
49 = 7 \times 7 = 7^2
\]
**Prime factorization of 42**:
\[
42 = 2 \times 3 \times 7
\]
Next, we identify the common prime factors and their lowest powers:
- The common prime factor is \(7\).
- The lowest power of \(7\) in the factorizations is \(7^1\).
Therefore, the greatest common divisor (GCD) is:
\[
GCD(49, 42) = 7
\]
This means the greatest number of tables that can be set up, with equal numbers of men's and women's shirts at each table, is \(7\).
To verify, we check how many shirts would be available at each table:
- Men's shirts per table:
\[
\frac{49}{7} = 7
\]
- Women's shirts per table:
\[
\frac{42}{7} = 6
\]
Thus, each table would have 7 men's shirts and 6 women's shirts, with no shirts left over.
Therefore, the greatest number of tables the band can sell shirts from is \(\boxed{7}\).
The numbers of T-shirts are:
- Men's T-shirts: 49
- Women's T-shirts: 42
First, we will find the prime factorization of both numbers.
**Prime factorization of 49**:
\[
49 = 7 \times 7 = 7^2
\]
**Prime factorization of 42**:
\[
42 = 2 \times 3 \times 7
\]
Next, we identify the common prime factors and their lowest powers:
- The common prime factor is \(7\).
- The lowest power of \(7\) in the factorizations is \(7^1\).
Therefore, the greatest common divisor (GCD) is:
\[
GCD(49, 42) = 7
\]
This means the greatest number of tables that can be set up, with equal numbers of men's and women's shirts at each table, is \(7\).
To verify, we check how many shirts would be available at each table:
- Men's shirts per table:
\[
\frac{49}{7} = 7
\]
- Women's shirts per table:
\[
\frac{42}{7} = 6
\]
Thus, each table would have 7 men's shirts and 6 women's shirts, with no shirts left over.
Therefore, the greatest number of tables the band can sell shirts from is \(\boxed{7}\).