# Thread: Math egg puzzle Help

1. ## Math egg puzzle Help

a peddler is taking edge to the market to sell. The eggs are in a cart that can hold 500 eggs. If the eggs are removed fom the cart 2 3 4 5 or 6 at a time 1 egg s alwas left over If the eggs are removed 7 at a time no eggs are left over how many eggs are their in the cart?

2. Hello, Dragon!

A peddler is taking eggs to the market to sell.
The eggs are in a cart that can hold 500 eggs.
If the eggs are removed fom the cart 2, 3, 4, 5 or 6 at a time,
. . one egg is always left over.
If the eggs are removed 7 at a time, no eggs are left over.
How many eggs are their in the cart?
Let $\displaystyle N$ be the number of eggs.

The LCM of 2, 3, 4, 5 and 6 is $\displaystyle 60$.
. . Hence, $\displaystyle 60$ (and any multiple of 60) is divisible by 2, 3, 4, 5 and 6.
To have a remainder of 1, $\displaystyle N$ must be of the form: .$\displaystyle 60k + 1$
. . for some positive integer $\displaystyle k$.
That is:$\displaystyle N \:=\:60k + 1$ [1]

Since $\displaystyle N$ is divisible by $\displaystyle 7\!:\;\;60k + 1 \:=\:7a$

If you're familiar with Modulo Arithmetic: .$\displaystyle 60k \,\equiv \,-1\!\! \pmod{7}$

Reduce: .$\displaystyle 4k \,\equiv \,6\!\! \pmod{7}$

Multiply by 2: .$\displaystyle 8k\,\equiv\,12\!\!\pmod{7}\quad\Rightarrow\quad k\,\equiv\,5\!\!\pmod{7}$

Hence, $\displaystyle k$ is of the form: $\displaystyle 7b + 5$ for some integer $\displaystyle b$.
That is: $\displaystyle k \:=\:7b + 5$ [2]

Substitute [2] into [1]: .$\displaystyle N \:=\:60(7b + 5) + 1\:=\:420b + 301$

Since $\displaystyle 0 < N \leq 500$, then $\displaystyle b = 0$

Therefore: .$\displaystyle \boxed{N \,= \,301}$

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If you're not familiar with Modulo Arithmetic,
. . it can be solved with "ordinary" algebra.
It just takes longer.