How many prime numbers exist between 200 and 220?

(A) None

(B) One

(C) Two

(D) Three

(E) Four

The OA is B.

Experts, what is the best approach that I could use to solve this PS question? May you give me some help?

## How many prime numbers exist between 200 and 220?

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Hello M7MBA.M7MBA wrote:How many prime numbers exist between 200 and 220?

(A) None

(B) One

(C) Two

(D) Three

(E) Four

The OA is B.

Experts, what is the best approach that I could use to solve this PS question? May you give me some help?

I would solve it like this:

We don't have to check the even numbers.

Now, from the odd numbers:

201, 207, 210, 213, 219 are divisible by 3 (because the sum of their digits is divisible by 3).

205, 215 are divisible by 5.

Hence, we have to check just the following numbers: 203, 209, 211 and 217. Now,

203 = 7*29 (NOT PRIME).

209 = 11*19 (NOT PRIME).

211 = PRIME.

217 = 7*31 (NOT PRIME).

Hence the correct answer is the option B.

One rule that can be useful is: if we want to know if x is a prime number, we have to divide it by all the primes between 2 and $$\sqrt{x}.$$ Since $$\sqrt{220}\approx14.8$$ then we have to check only by 2, 3, 5, 7, 11 and 13.

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Vincen has an excellent explanation here. The one thing I'll add is that if you find it difficult to see that 209 is divisible by 11, you can start with a number in the neighborhood that we know is divisible by 11, such as 220, and extrapolate. If 220 = 11*20, then 209, or 220 - 11, would contain one fewer 11.Vincen wrote:Hello M7MBA.M7MBA wrote:How many prime numbers exist between 200 and 220?

(A) None

(B) One

(C) Two

(D) Three

(E) Four

The OA is B.

Experts, what is the best approach that I could use to solve this PS question? May you give me some help?

I would solve it like this:

We don't have to check the even numbers.

Now, from the odd numbers:

201, 207, 210, 213, 219 are divisible by 3 (because the sum of their digits is divisible by 3).

205, 215 are divisible by 5.

Hence, we have to check just the following numbers: 203, 209, 211 and 217. Now,

203 = 7*29 (NOT PRIME).

209 = 11*19 (NOT PRIME).

211 = PRIME.

217 = 7*31 (NOT PRIME).

Hence the correct answer is the option B.

One rule that can be useful is: if we want to know if x is a prime number, we have to divide it by all the primes between 2 and $$\sqrt{x}.$$ Since $$\sqrt{220}\approx14.8$$ then we have to check only by 2, 3, 5, 7, 11 and 13.

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First, we can omit all the even numbers (since they are divisible by 2) and all the odd numbers ending in 5 (since they are divisible by 5). So we are left with 201, 203, 207, 209, 211, 213, 217 and 219. We can omit 201, 207, 213 and 219 also since all of these numbers are divisible by 3 (notice that the sum of their digits is divisible by 3). So we only need to consider 203, 209, 211 and 217.M7MBA wrote:How many prime numbers exist between 200 and 220?

(A) None

(B) One

(C) Two

(D) Three

(E) Four

The OA is B.

203/7 = 29 â†’ So 203 is not a prime.

209/7 = 29 R 6, 209/11 = 19 â†’ So 209 is not a prime.

211/7 = 30 R 1, 211/11 = 19 R 2, 211/13 = 16 R 3, 211/17 = 12 R 7 â†’ So 211 is a prime.

217/7 = 31.â†’ So 217 is not a prime.

Therefore, there is only 1 prime number between 200 and 220.

Answer: B

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