Deciding whether a number is prime

Statement

Decide whether each number is prime, testing divisibility by the primes up to its square root.

(i) $103$ (ii) $217$ (iii) $323$ (iv) $149$

Solution

(i) $\sqrt{103} \approx 10.1$: test the primes $2,3,5,7$. None divides $103$, so $103$ is prime. (ii) $217 = 7 \cdot 31$, so not prime. (iii) $323 = 17 \cdot 19$, so not prime. (iv) $\sqrt{149} \approx 12.2$: test $2,3,5,7,11$. None divides $149$, so $149$ is prime.