Question
At a family reunion, each of Lacey's aunts and uncles is getting photographed once. The aunts are taking pictures in groups of 10 and the uncles are taking pictures in groups of 5. If Lacey has the same total number of aunts and uncles, what is the minimum number of aunts that Lacey must have?
Answers
GPT-4o mini
Let \( a \) be the number of aunts Lacey has and \( u \) be the number of uncles. According to the problem, we have the following conditions:
1. The aunts are photographed in groups of 10.
2. The uncles are photographed in groups of 5.
3. The total number of aunts equals the total number of uncles, i.e., \( a = u \).
The total number of groups of aunts is given by \( \frac{a}{10} \) and for uncles, it is given by \( \frac{u}{5} \).
Since \( a = u \), we can replace \( u \) with \( a \) in the equation for uncles:
\[
\text{Number of uncle groups} = \frac{u}{5} = \frac{a}{5}.
\]
The problem states that both \( \frac{a}{10} \) and \( \frac{a}{5} \) should result in whole numbers, implying that both \( a \) must be divisible by 10 and 5.
To satisfy the condition for groups of aunts,
\[
a \text{ must be divisible by } 10.
\]
Given the divisibility by 5 (which is inherently satisfied since 10 is also divisible by 5), we conclude that \( a \) must be a multiple of 10.
The smallest positive multiple of 10 is \( 10 \). Thus, the minimum number of aunts that Lacey must have is:
\[
\boxed{10}.
\]
To verify, if Lacey has 10 aunts, then she must also have \( 10 \) uncles (since \( a = u \)). Hence, for the aunts:
- Number of groups of aunts = \( \frac{10}{10} = 1 \).
And for the uncles:
- Number of groups of uncles = \( \frac{10}{5} = 2 \).
Both calculations yield whole numbers confirming that Lacey having 10 aunts and 10 uncles works perfectly within the conditions provided.
1. The aunts are photographed in groups of 10.
2. The uncles are photographed in groups of 5.
3. The total number of aunts equals the total number of uncles, i.e., \( a = u \).
The total number of groups of aunts is given by \( \frac{a}{10} \) and for uncles, it is given by \( \frac{u}{5} \).
Since \( a = u \), we can replace \( u \) with \( a \) in the equation for uncles:
\[
\text{Number of uncle groups} = \frac{u}{5} = \frac{a}{5}.
\]
The problem states that both \( \frac{a}{10} \) and \( \frac{a}{5} \) should result in whole numbers, implying that both \( a \) must be divisible by 10 and 5.
To satisfy the condition for groups of aunts,
\[
a \text{ must be divisible by } 10.
\]
Given the divisibility by 5 (which is inherently satisfied since 10 is also divisible by 5), we conclude that \( a \) must be a multiple of 10.
The smallest positive multiple of 10 is \( 10 \). Thus, the minimum number of aunts that Lacey must have is:
\[
\boxed{10}.
\]
To verify, if Lacey has 10 aunts, then she must also have \( 10 \) uncles (since \( a = u \)). Hence, for the aunts:
- Number of groups of aunts = \( \frac{10}{10} = 1 \).
And for the uncles:
- Number of groups of uncles = \( \frac{10}{5} = 2 \).
Both calculations yield whole numbers confirming that Lacey having 10 aunts and 10 uncles works perfectly within the conditions provided.