Find the smallest number of eggs

[Math100]

Homework Statement

(Brahmagupta, 7th Century A.D.) When eggs in a basket are removed $2, 3, 4, 5, 6$ at a time there remain, respectively, $1, 2, 3, 4, 5$ eggs. When they are taken out $7$ at a time, none are left over. Find the smallest number of eggs that could have been contained in the basket.

Relevant Equations

None.

Let $x$ be the smallest number of eggs.
Then
\begin{align*}
&x\equiv -1\pmod {2}\equiv 1\pmod {2}\\
&x\equiv -1\pmod {3}\equiv 2\pmod {3}\\
&x\equiv -1\pmod {4}\equiv 3\pmod {4}\\
&x\equiv -1\pmod {5}\equiv 4\pmod {5}\\
&x\equiv -1\pmod {6}\equiv 5\pmod {6}\\
&x\equiv 0\pmod {7}.\\
\end{align*}
Note that $lcm(2, 3, 4, 5, 6)=60$.
This means $x\equiv -1\pmod {60}\equiv 59\pmod {60}$.
Now we have $x=59+60m$ for some $m\in\mathbb{N}$.
Thus $x=59+60(1)=119\implies x\equiv 0\pmod {7}$.
Therefore, the smallest number of eggs that could have been contained in the basket is $119$.

 

[fresh_42]

Homework Statement:: (Brahmagupta, 7th Century A.D.) When eggs in a basket are removed $2, 3, 4, 5, 6$ at a time there remain, respectively, $1, 2, 3, 4, 5$ eggs. When they are taken out $7$ at a time, none are left over. Find the smallest number of eggs that could have been contained in the basket.
Relevant Equations:: None.

Let $x$ be the smallest number of eggs.
Then
\begin{align*}
&x\equiv -1\pmod {2}\equiv 1\pmod {2}\\
&x\equiv -1\pmod {3}\equiv 2\pmod {3}\\
&x\equiv -1\pmod {4}\equiv 3\pmod {4}\\
&x\equiv -1\pmod {5}\equiv 4\pmod {5}\\
&x\equiv -1\pmod {6}\equiv 5\pmod {6}\\
&x\equiv 0\pmod {7}.\\
\end{align*}
Note that $lcm(2, 3, 4, 5, 6)=60$.
This means $x\equiv -1\pmod {60}\equiv 59\pmod {60}$.
Now we have $x=59+60m$ for some $m\in\mathbb{N}$.
Thus $x=59+60(1)=119\implies x\equiv 0\pmod {7}$.
Therefore, the smallest number of eggs that could have been contained in the basket is $119$.

Correct.

I didn't know that Brahmagupta had more problems associated with him. I only knew a geometric problem:
problem 13 in
https://www.physicsforums.com/threads/math-challenge-september-2019.976793/
solution on page 380f. in the solution manual (last attachment)
https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/