This is a homework problem that I'm confused about. I understand the solution until it says "With the Chinese Remainder Theorem and some computation this shows that $n \equiv 301 \pmod{420}.$ I'm not sure how to use Chinese Remainder Theorem and how to get this, any explanation is appreciated.
A woman with a basket of eggs finds that if she removes either 2, 3, 4, 5, or 6 at a time from the basket, there is always one egg left over. If she removes 7 eggs at a time from the basket, there are no eggs left over. If the basket holds up to 500 eggs, how many eggs does she have?
Solution.
Let $n$ be the number of eggs in the basket. We know that $n$ is $1$ mod $2,3,4,5,6$. This means $n-1$ is $0$ mod $2,3,4,5,6$ and is divisible by all of those numbers. Since the LCM of $2,3,4,5,6$ is $60$, $n-1$ is a multiple of $60$, and $n \equiv 1 \pmod{60}$. Finally, we know that $n \equiv 0 \pmod{7}$. With the Chinese Remainder Theorem and some computation this shows that $n \equiv 301 \pmod{420}$. So the possible values of $n$ are $301, 721, 1141, \ldots$ and the only one less than $500$ is $\boxed{301}$
Best Answer
You can just try the positive integers that are congruent to $1$ modulo $60$ until you find one that is also divisible by $7.$ The first few $61,121,181,241$ don't work, but $301=7\cdot 43$ does work. I believe the function of CRT here is to let you know that this is the unique solution that is a positive integer that is less than or equal to $60\cdot 7=420.$