How many prime numbers are there between 200 and 230?

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UPSC CISF-AC-EXE – 2019

To find the number of prime numbers between 200 and 230, we need to check each integer in this range (from 201 to 229) for primality. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To check if a number ‘n’ is prime, we can test for divisibility by prime numbers up to the square root of ‘n’. The square root of 230 is approximately 15.17. So, we need to check for divisibility by primes up to 13 (2, 3, 5, 7, 11, 13).

Let’s check the numbers between 200 and 230:
– 201: Divisible by 3 (sum of digits 2+0+1=3). Not prime.
– 202: Divisible by 2. Not prime.
– 203: $203 = 7 \times 29$. Not prime.
– 204: Divisible by 2. Not prime.
– 205: Divisible by 5. Not prime.
– 206: Divisible by 2. Not prime.
– 207: Divisible by 3 (sum of digits 2+0+7=9). Not prime.
– 208: Divisible by 2. Not prime.
– 209: $209 = 11 \times 19$. Not prime.
– 210: Divisible by 10. Not prime.
– 211: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $14^2 = 196$, $15^2 = 225$. $\sqrt{211} \approx 14.5$. Primes to check up to 13. 211 is prime.
– 212: Divisible by 2. Not prime.
– 213: Divisible by 3 (sum of digits 2+1+3=6). Not prime.
– 214: Divisible by 2. Not prime.
– 215: Divisible by 5. Not prime.
– 216: Divisible by 2. Not prime.
– 217: $217 = 7 \times 31$. Not prime.
– 218: Divisible by 2. Not prime.
– 219: Divisible by 3 (sum of digits 2+1+9=12). Not prime.
– 220: Divisible by 10. Not prime.
– 221: $221 = 13 \times 17$. Not prime.
– 222: Divisible by 2. Not prime.
– 223: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{223} \approx 14.9$. Primes to check up to 13. 223 is prime.
– 224: Divisible by 2. Not prime.
– 225: Divisible by 5. Not prime.
– 226: Divisible by 2. Not prime.
– 227: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{227} \approx 15.07$. Primes to check up to 13. 227 is prime.
– 228: Divisible by 2. Not prime.
– 229: Check divisibility by primes 2, 3, 5, 7, 11, 13. Not divisible by any of these. $\sqrt{229} \approx 15.13$. Primes to check up to 13. 229 is prime.

The prime numbers between 200 and 230 are 211, 223, 227, and 229.
There are 4 prime numbers in this range.

– A prime number is a natural number greater than 1 with no positive divisors other than 1 and itself.
– To check for primality of a number ‘n’, one needs to test divisibility only by prime numbers up to $\sqrt{n}$.

Checking primality efficiently requires testing only prime divisors. For numbers up to 230, the largest prime divisor we need to check is 13, since the next prime is 17, and $17^2 = 289 > 230$.
The prime numbers less than or equal to 13 are 2, 3, 5, 7, 11, 13.

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