What is the largest 3-digit prime number?

The answer is 997.

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Let's check from the top. Since $999 = 3 \times 333$ and $998 = 2 \times 449$ , neither of them will be prime. We will show that $997$ is prime.

To check that a number $N$ is prime, we just have to check that none of the integers from $2$ to $\sqrt{N}$ are a divisor of $N$ . This is because if $N= a \times b$ , then one of the numbers $a$ or $b$ must be at most $\sqrt{N}$ .

Moreover, it suffices to only check prime numbers. Since $\sqrt{ 997} < 32$ , let's do the division.

$\begin{aligned} 997 & =\ 2 \times 498\ +\ 1 \\ 997 & =\ 3 \times 332\ +\ 1 \\ 997 & =\ 5 \times 199\ +\ 2 \\ 997 & =\ 7 \times 142\ +\ 3 \\ 997 & =\ 11 \times 90\ +\ 7 \\ 997 & =\ 13 \times 76\ +\ 9 \\ 997 & =\ 17 \times 58\ +\ 11 \\ 997 & =\ 23 \times 43\ +\ 8 \\ 997 & =\ 29 \times 34\ +\ 11 \\ 997 & =\ 31 \times 32\ +\ 5 \\ \end{aligned}$

Hence, the above shows that 997 is indeed prime. Thus, it is the largest 3-digit prime number.