# Find the smallest number of eggs

• Math100
In summary, the conversation discusses a problem posed by Brahmagupta in the 7th century involving eggs in a basket being removed in certain quantities. Using modular arithmetic and the least common multiple, it is determined that the smallest number of eggs that could have been in the basket is 119. This is different from the geometric problem associated with Brahmagupta that was previously known.
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 ##.

Math100 said:
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
solution on page 380f. in the solution manual (last attachment)

Math100

## What is the purpose of finding the smallest number of eggs?

The purpose of finding the smallest number of eggs is to determine the minimum number of eggs needed in a given scenario. This can be useful in various situations, such as calculating the minimum amount of ingredients needed for a recipe or determining the minimum number of eggs that need to be purchased for a specific event.

## How do you find the smallest number of eggs?

To find the smallest number of eggs, you can use basic mathematical principles such as division and multiplication. First, determine the total number of eggs needed and then divide it by the number of eggs in each carton. The resulting number will be the minimum number of cartons needed. Then, multiply this number by the number of eggs in each carton to get the smallest number of eggs needed.

## What factors should be considered when finding the smallest number of eggs?

When finding the smallest number of eggs, it is important to consider factors such as the number of people or servings, the recipe or event requirements, and the size of the egg cartons available. Additionally, any potential waste or leftovers should also be taken into account.

## Can the smallest number of eggs change depending on the situation?

Yes, the smallest number of eggs can change depending on the situation. Factors such as the number of people or servings, the recipe or event requirements, and the size of the egg cartons available can all affect the minimum number of eggs needed. Therefore, it is important to reassess and recalculate the smallest number of eggs for each unique scenario.

## Why is it important to find the smallest number of eggs?

Finding the smallest number of eggs is important because it can help save time, money, and resources. By determining the minimum number of eggs needed, you can avoid purchasing excess eggs that may go to waste. This can also help with meal planning and budgeting. Additionally, finding the smallest number of eggs can also be a fun and challenging mathematical exercise.

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