Question
Independent: There is a caravan of camels. The camels walk in different formations. Whether the camels walk in rows of 2, 3, 4, or 5, one camel walks alone. What is the least number of camels that could be in the caravan?
Answers
GPT-4o mini
Let the number of camels in the caravan be denoted by \( N \). According to the problem, when the camels walk in rows of 2, 3, 4, or 5, there is always one camel that remains alone. This indicates the following congruences:
\[
N \equiv 1 \mod 2
\]
\[
N \equiv 1 \mod 3
\]
\[
N \equiv 1 \mod 4
\]
\[
N \equiv 1 \mod 5
\]
This means that \( N - 1 \) must be divisible by 2, 3, 4, and 5. In other words, we can express this as:
\[
N - 1 = k \quad \text{for some integer \( k \)},
\]
where \( k \) is a common multiple of 2, 3, 4, and 5.
To find the least value of \( k \), we need to determine the least common multiple (LCM) of these numbers.
Calculating the LCM:
- The prime factorization of each number is:
- \( 2 = 2^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
To find the LCM, we take the highest power of each prime that appears:
- For the prime \( 2 \), the highest power is \( 2^2 \) from 4.
- For the prime \( 3 \), the highest power is \( 3^1 \).
- For the prime \( 5 \), the highest power is \( 5^1 \).
Thus, the LCM is calculated as follows:
\[
\text{LCM}(2, 3, 4, 5) = 2^2 \times 3^1 \times 5^1
\]
Calculating it step by step:
\[
2^2 = 4,
\]
\[
4 \times 3 = 12,
\]
\[
12 \times 5 = 60.
\]
So, \( k = 60 \). Since \( N - 1 = k \), we have:
\[
N - 1 = 60 \quad \Rightarrow \quad N = 60 + 1 = 61.
\]
Finally, we check the conditions:
- For rows of 2: \( 61 \div 2 = 30\) remainder 1.
- For rows of 3: \( 61 \div 3 = 20\) remainder 1.
- For rows of 4: \( 61 \div 4 = 15\) remainder 1.
- For rows of 5: \( 61 \div 5 = 12\) remainder 1.
Since all conditions are satisfied, the least number of camels that could be in the caravan is:
\[
\boxed{61}.
\]
\[
N \equiv 1 \mod 2
\]
\[
N \equiv 1 \mod 3
\]
\[
N \equiv 1 \mod 4
\]
\[
N \equiv 1 \mod 5
\]
This means that \( N - 1 \) must be divisible by 2, 3, 4, and 5. In other words, we can express this as:
\[
N - 1 = k \quad \text{for some integer \( k \)},
\]
where \( k \) is a common multiple of 2, 3, 4, and 5.
To find the least value of \( k \), we need to determine the least common multiple (LCM) of these numbers.
Calculating the LCM:
- The prime factorization of each number is:
- \( 2 = 2^1 \)
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
To find the LCM, we take the highest power of each prime that appears:
- For the prime \( 2 \), the highest power is \( 2^2 \) from 4.
- For the prime \( 3 \), the highest power is \( 3^1 \).
- For the prime \( 5 \), the highest power is \( 5^1 \).
Thus, the LCM is calculated as follows:
\[
\text{LCM}(2, 3, 4, 5) = 2^2 \times 3^1 \times 5^1
\]
Calculating it step by step:
\[
2^2 = 4,
\]
\[
4 \times 3 = 12,
\]
\[
12 \times 5 = 60.
\]
So, \( k = 60 \). Since \( N - 1 = k \), we have:
\[
N - 1 = 60 \quad \Rightarrow \quad N = 60 + 1 = 61.
\]
Finally, we check the conditions:
- For rows of 2: \( 61 \div 2 = 30\) remainder 1.
- For rows of 3: \( 61 \div 3 = 20\) remainder 1.
- For rows of 4: \( 61 \div 4 = 15\) remainder 1.
- For rows of 5: \( 61 \div 5 = 12\) remainder 1.
Since all conditions are satisfied, the least number of camels that could be in the caravan is:
\[
\boxed{61}.
\]