What is the number that, when divided by $3$, $5$, $7$, leaves remainders of $2$, $3$, $2$, respectively? What is the number? The LCM of the divisors is 105. I think this has something to do with the Chinese remainer theorem, but I am not sure how to apply this knowledge.

Yes, this is the first known incidence of the **Chinese Remainder Theorem** (CRT).

From Wikipedia:

> The earliest known statement of the CRT, as a problem with specific numbers, appears in the 3rd-century book Sunzi's Mathematical Classic () by the Chinese mathematician Sun Tzu:[2] “ There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there?"

The solution given by the CRT algorithm is indeed $x\equiv 23 \bmod 105=3\cdot 5\cdot 7$, as shown there. So there are $23+k\cdot 105$ things, for $k\in \mathbb{N}$.