An egg man was walking with eggs in a basket, unexpectedly a person coming in the opposite direction on a bicycle bumped into the egg man, all the eggs were broken, the egg man bought all the broken eggs and the egg man accounted for it....The egg maker calculates that if you divide by 2 you get 1 remainder, if you divide by 3 you get 1 remainder, if you divide by 4 you get 1 remainder, if you divide by 5 you get 1 remainder, if you divide by 6 you get 1 remainder, if you divide by 7 there is no remainder that is the number of eggs

This problem is a classic example of finding a number that satisfies a set of modular conditions. We need to find the smallest positive integer \( x \) that satisfies the following conditions:

\[
\begin{align*}
x &\equiv 1 \pmod{2}, \\
x &\equiv 1 \pmod{3}, \\
x &\equiv 1 \pmod{4}, \\
x &\equiv 1 \pmod{5}, \\
x &\equiv 1 \pmod{6}, \\
x &\equiv 0 \pmod{7}.
\end{align*}
\]

### Step 1: Analyzing the Modular Conditions
The first five conditions tell us that \( x \) leaves a remainder of 1 when divided by 2, 3, 4, 5, or 6. This implies that \( x - 1 \) is a multiple of the least common multiple (LCM) of 2, 3, 4, 5, and 6.

Let's calculate the LCM:

\[
\text{LCM}(2, 3, 4, 5, 6) = \text{LCM}(60, 4) = 60
\]

Thus, \( x - 1 \) is a multiple of 60. We can express \( x \) as:

\[
x = 60k + 1 \quad \text{for some integer } k.
\]

### Step 2: Applying the Last Condition
The last condition tells us that \( x \) is divisible by 7, i.e.,

\[
x \equiv 0 \pmod{7}.
\]

So, we need:

\[
60k + 1 \equiv 0 \pmod{7}.
\]

This simplifies to:

\[
60k \equiv -1 \pmod{7}.
\]

Since \( 60 \equiv 4 \pmod{7} \), we have:

\[
4k \equiv -1 \pmod{7} \quad \text{or equivalently} \quad 4k \equiv 6 \pmod{7}.
\]

To solve this, multiply both sides by the modular inverse of 4 modulo 7. The inverse of 4 modulo 7 is 2, so:

\[
k \equiv 6 \times 2 \pmod{7} \quad \Rightarrow \quad k \equiv 12 \pmod{7} \quad \Rightarrow \quad k \equiv 5 \pmod{7}.
\]

### Step 3: Finding the Smallest \( x \)
Substitute \( k = 7m + 5 \) into \( x = 60k + 1 \):

\[
x = 60(7m + 5) + 1 = 420m + 301.
\]

The smallest positive \( x \) occurs when \( m = 0 \):

\[
x = 301.
\]

Thus, the smallest number of eggs that satisfies all the conditions is **301**.

Do you have any questions or need further details? 

Here are 5 related questions:
1. How is the least common multiple (LCM) of a set of numbers calculated?
2. What is the Chinese Remainder Theorem, and how does it relate to problems like this?
3. Can you find other numbers that satisfy the given conditions?
4. How do you calculate modular inverses, and why are they useful in solving such problems?
5. What is the significance of the number 301 in this context?

**Tip:** Modular arithmetic is a key tool in number theory and is widely used in areas like cryptography, coding theory, and solving mathematical puzzles.

Explore how to solve a modular arithmetic problem involving eggs: finding the smallest number that satisfies multiple modular conditions. This problem integrates concepts like modular arithmetic, the Chinese Remainder Theorem, and number theory. Suitable for advanced high school students and beyond.

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