Question
In preparation for a conference, Nicole is setting up some stations where people can create their own name tags. She has 14 name tags and 18 pens, which she wants to distribute evenly among the name tag stations with none left over. What is the greatest number of name tag stations that Nicole can set up?
Answers
GPT-4o mini
To find the greatest number of name tag stations Nicole can set up, we need to determine the largest number that evenly divides both the total number of name tags (14) and the total number of pens (18).
We will start by finding the greatest common divisor (GCD) of the two numbers:
1. **Prime factorization**:
- The prime factorization of 14 is \( 2 \times 7 \).
- The prime factorization of 18 is \( 2 \times 3^2 \).
2. **Finding GCD**:
- The common prime factor between 14 and 18 is 2.
- The minimum power of the common factor is \( 2^1 \).
Thus, the GCD of 14 and 18 is \( 2 \).
3. **Conclusion**:
- Therefore, the greatest number of name tag stations that Nicole can set up is \( \boxed{2} \).
To verify, we can check how both 14 and 18 divide by 2:
- Dividing the name tags: \( 14 \div 2 = 7 \) name tags per station.
- Dividing the pens: \( 18 \div 2 = 9 \) pens per station.
Since both divisions are whole numbers with no remainder, our answer is confirmed. The greatest number of name tag stations is indeed \( \boxed{2} \).
We will start by finding the greatest common divisor (GCD) of the two numbers:
1. **Prime factorization**:
- The prime factorization of 14 is \( 2 \times 7 \).
- The prime factorization of 18 is \( 2 \times 3^2 \).
2. **Finding GCD**:
- The common prime factor between 14 and 18 is 2.
- The minimum power of the common factor is \( 2^1 \).
Thus, the GCD of 14 and 18 is \( 2 \).
3. **Conclusion**:
- Therefore, the greatest number of name tag stations that Nicole can set up is \( \boxed{2} \).
To verify, we can check how both 14 and 18 divide by 2:
- Dividing the name tags: \( 14 \div 2 = 7 \) name tags per station.
- Dividing the pens: \( 18 \div 2 = 9 \) pens per station.
Since both divisions are whole numbers with no remainder, our answer is confirmed. The greatest number of name tag stations is indeed \( \boxed{2} \).