How many integers N are prime numbers in the range 200 < N < 220?

So, the easiest way to reduce the calculation in finding a prime number is this:

To test a number N on whether it is a prime number , take square root(N) .
Consider as it is if it is a natural number. Otherwise, increase the sq. root to the next natural no.
Then divide given no. by all prime numbers BELOW the sq. root obtained.
If the no. is divisible by any of the prime numbers, then it is not a prime number; else it is.

Square root of 200 or 220 is less than 15
So, lets check if any number between 200 and 220 is not divisible by prime numbers 2,3,5,7,11(<15). That's it.

Out: 201, 202, 203, 204, 205, 206, 207, 208, 209, 210,
212, 213, 214, 215, 216, 217, 218, 219

Only 211 is a prime number in the given range
Correct Option : A

Another point about prime numbers is that every prime number >3 can be written in the form of \(6n \pm 1\) which is what Abhishek009 has used above.

Dear Yosita,
I'm happy to respond.

My friend, the 6n + 3 is a typo, a misprint. As I am sure you appreciate, we all make mistakes. If n is a positive integer, then any number of the form (6n + 3) = 3(2n + 1) would automatically be divisible by 3 and therefore NEVER would be prime. Beyond the single digits primes (for which you should need no rule!), all prime numbers are of the forms (6n + 1) or (6n - 1).
Not all numbers of that form are prime, but all primes are of that form.

Actually, I think this formula rule is almost completely useless. Suppose we want to find the prime numbers between 150 and 160. First of all, we would automatically eliminate all the even numbers and multiples of 5, which leaves us with {151, 153, 157, 159}. Then we eliminate the multiples of three, 153 and 159, so that we are left with {151, 157}. These two are the remaining candidates for prime numbers, because they are odd numbers not divisible by three. That's all the formula thing does for you: it gets you odd numbers not divisible by three, but we can get to that point much more easily without touching the formula. Now, are those two number prime? Look at the multiples of the next few prime numbers.

We know 14 is a multiple of 7, so 140 has to be a multiple of 7 as well. Thus, 140 + 7 = 147, 147 + 7 = 154, and 154 + 7 = 161 are multiples of 7. Thus, 151 and 157 are not multiple of 7 or anything less than 7.

We know 110 is a multiple of 11, so we add 11 to get more: 121, 132, 143, 154, 165, etc. Thus, 151 and 157 are not multiples of 11 or anything less than 11.

We know 130 is a multiple of 13, so we add 13 to get more: 143, 156, 169, etc. Thus, 151 and 157 are not multiples of 13 or anything less than 13.

Well, we only need check up the square root of a number. Since 13^2 = 169, we know the square roots of 151 and 157 are less than 13. If those two numbers are not divisible by anything less than 13, they are are not divisible by anything greater than 13. Thus, 151 and 157 are prime numbers.

It's much easier to ignore that (6n + 1) or (6n - 1) rule entirely and just check the numbers direct. That formula simple gets in the way.

Does all this make sense?
Mike

I dont think I have seen any other property of prime numbers getting applied in official questions.

Hmmm....

I got it where U are having problem !!


1. Mike has stated that every Prime Number (except 2) is a ODD number ( As we all Know )

2. Every Odd number can be found out using the formula 4n + 1 & 4n + 3

3. Every Prime number greater than 2 can be found out using the formula 6n + 1 & 6n - 3


Find set of odd numbers = { 5 , 7 , 9 , 11 , 13 , 15..........}

The red Odd numbers are not Prime numbers ...

No calculate 6n + 1 and 6n - 1 rule

Find set of Prime numbers = { 5 , 7 , 11 ,13 .......}

So he has stated -



Actually Prime Numbers are subsets of ODD numbers as represented below -



So, if you try to come to prime numbers using ODD number formula of 4n + 1 & 4n + 3 then you will find some numbers which are non Prime.

Hope I am clear this time..

Fab Work.
I think i get it now
I added the same question from E-gmat forum too.
if-n-is-a-positive-integer-which-of-the-following-must-be-true-217202.html


Reagards
StoneCold

Great minds think alike. Veritas already had this exact question: how-many-prime-numbers-exist-between-200-and-185996.html

mike

Hallo Mike,being a premium subscriber of Magoosh and a follower of your blog,I know that you have a common interest in history and facts.I would like to share a very interesting fact regarding this question.The famous mathematician G.H Hardy,who was also the mentor of famous mathematician Ramanujan, has made this particular problem very easy for any mathematics student.

Once he was asked that what he would like to accomplish in his life apart from mathematics.He said that he wanted to score 211 runs at Oval against Australia in test cricket.When asked why 211,he answered :"because its the first prime after 200 dear",thus making 211 eternal for any math enthusiastic.So any mathematics student would know that between 200 and 220 the only prime is 211....

This is a very good question Mike,but I found it very funny....

Very interesting solution Abhishek009
kudos....!
Fun fact => every integer (>5) which is prime can also be written as either 4N+1 or 4N+3

here the rule i used is well I didnt use any rule .
i checked every value for divisibility with prime numbers less than 15 (given that 15^2=225 and all values are less than 225 and above 200
It did not take longer than 2 minutes.
To find if a number is prime or not we need to check the divisibility with prime numbers less than the square root of that number.



Excellent Question mike ..!!!

looking forward to your solution too mikemcgarry

Great..
IS there any more properties like these two?

regards
StoneCold

Thanks for the response mike.
Excellent Solution and i will be using this in the future questions

I however disagree with the statement regarding 4N+1 and 4N-1
i believe every prime number that exists on the planet can be written either as 4N+1 or as 4N-1
Let me know if i am missing something.


Regards
StoneCold

Hey Stunnerman !!

Plz check the highlighted part of your post....

Every ODD Number can be represented as 4n + 1 or 4n + 3 ( If you need to can prove the same algebraically )

Mike has stated -




Hope its clear now !!

Actually I still don't get it
I saw this on the E-gmat's portal that every prime can be written either as 4N+1 or as 4N-1
Is it correct or not?
I am not talking about odds or evens here
Just in general for primes.

Regards
StoneCold

Dear Bunuel,

OOOPS! Thank you very much, Bunuel, for point this out! Apologies to Veritas! I had no idea!

Mike

Dear Stone Cold,

I see that Abhishek009 gave you a very thoughtful response. I will just add my thoughts.

It is 100% true that every prime number greater than 2 can be written as 4N-1 or 4N+1. This is a true but useless statement. Consider these statements:
1) All world leaders and heads of state around the world are either males or females.
2) All the illegal narcotics traded around the world are substances other than water and sugar.
3) All the Veritas SC and CR questions are five-choice multiple choice questions with only one right answer.

All of those statements are 100% true, and they are absolutely useless! Similarly, the statement that every prime number greater than 2 can be written as 4N-1 or 4N+1. This is simply the algebraic statement that every prime number greater that 2 is odd. Again, 100% true but it doesn't help you find anything. It is a true statement that contains no useful information. Don't get stuck on the fact that it is true: of course it is true, but that doesn't help us at all.

If we want to analyze any stretch of numbers greater than 100 and find prime numbers, we absolute know that none of the even numbers in that range would prime. Since we know that, this algebraic statement adds no additional information.

There is a big difference between true & useless vs. simply technically true.

Does this make more sense now?
Mike

[quote]
Abhishek:
Nice analogy.
Please let me know at one time, we say that
Prime Number > 3 can be represented as "6n+1" or "6n-1".
Where,
The 6N+1 or 6N-1 rule is basically every odd number that is not divisible by three,
so it narrows the search a little.

Then,
Why do we represent Prime Number as "6n+1" or "6n-3"?
Why are we representing "6n -1" at one place and "6n-3" at another place?
Hope I am clear in my question.
Thanks in advance.

Regards,
Yosita

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