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سرفصل: بخش ریاضی / سرفصل: خواص عدد صحیح / درس 4

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## Prime Numbers

Prime numbers, in some sense, the most important positive integers are the prime numbers. Prime numbers are the building blocks of all other positive integers. Before we can discuss how powerful prime numbers are, we need to discuss simply what they are. What exactly is a prime number?

Recall first of all, that 1 Is a factor of every single positive integer. Recall also that every positive integer is a factor of itself. 3 is a factor of 3, 7 is a factor of 7. Thus, every positive integer has at least these two factors. Now of course many integers have many more factors. And in fact in a previous video we talked about counting the number of factors for large numbers.

A prime number has only these two factors, 1 and itself, and absolutely no other factors. In other words, a prime number is not divisible by any number lower than itself other than 1. So that’s the fundamental definition of a prime number. Here are the prime numbers less than 20.

You definitely need to know that those eight numbers are prime. You definitely need to recognize them instantly as prime numbers. That’s very important. And in fact, I’ll say here are the ones between 20 and 60. It would be a very good idea to know each one of these is prime as well. You don’t want to get stuck on the test having to figure out, now wait a second.

Is 51 a prime number? Is 53 a prime number? You don’t want to have to do those calculations in the middle of the test. It’s better just to know that these are the prime numbers. Above 60, you rarely need to know whether those numbers are prime. It’s easy enough to figure out the few instances where you would need to know that.

If you’re gung ho about memorizing primes, memorize them all the way up to 100, but you don’t need to do so. Here’s some important prime facts. Notice first of all, 1 is not a prime number. Prime numbers have two factors, 1 and the number itself, but 1 only has one factor, itself.

Now, I know some people get almost sentimental about this and they feel like 1 is being cheated or somehow it’s being artificially excluded from the elite club of prime numbers. So I just want to assure you, yes, mathematicians could have bent the rules, they could have made 1 a prime number if they wanted.

There are actually very sound mathematical reasons why mathematicians have chosen to define it this way, so that 1 is not a prime number. And in fact if you’re interested in some of these reasons, I highly suggest underneath this video, there will be a link to a GMAT blog about this very topic, and I’ll go into that in a little more detail in that blog.

But the important fact to recognize here, 1 is definitely not a prime number. Notice also, 2 is the only even prime number. All other even numbers that are divisible by 2, and hence are not prime. So, every prime number in the world is odd, except for 2. 2 is both the lowest prime number, and the only even prime number. The test absolutely loves to test these two facts.

I would say more than 50% of the time when they’re asking questions about prime numbers, at least one of these two facts is hidden somewhere in the question, they’re extremely important. Now we can talk a little about testing whether a larger number is prime. To test whether any number is less than 100 is prime, all we have to do is check whether it is divisible by one of the prime numbers less than 10.

If a number less than 100 is not divisible by any prime divisor less than 10, then the number has to be prime. And, of course, the only prime divisors less than 10 are 2, 3, 5, and 7. So those are the only divisors we have to check. It’s not divisible by any of those numbers, it is a prime number. So for example, here’s a practice problem.

I’m gonna say pause the video and work on this problem now. Okay, for all the positive integers N such that N is somewhere between 80 and 90, how many prime numbers are there? Well, fact number one, we know obviously none of the even numbers can be prime, cuz they were all divisible by 2.

So we’re only looking at the odd numbers. So here are the odd numbers in that sequence. Obviously 85 ends in a 5. It’s divisible by 5, so we can eliminate that. Well now you may know that 81 is 9 squared. So it’s divisible by 3, it’s divisible by 9.

So that’s obviously not a prime number. If we use our divisibility tricks we can check for divisibility by 3. 8 plus 3 is 11, not divisible by 3. 8 plus 7, 15, that is divisible by 3. 8 plus 9, 17 not divisible by 3. So 87.

The digits are divisible by 3 when we add up the sum of the digits. So that means that 87 is also divisible by 3. Another way to see that incidentally, obviously 90 is a multiple of 3. 90 minus 3 is 87. So, that means that 87 has to be a multiple of 3. So, we can eliminate 87.

And that leaves us with 83 and 89. So, we’ve already checked 2, 5, and 3. That leaves us with 7. Well, think about this. 7 obviously goes into 77. Add 7, it goes into 84.

Add 7, it goes into 91. Those are the multiples of 7 through this region. And the multiples of 7 pass through the 80s, they land on the even number 84, they don’t hit any of the odd numbers in the 80s. So 83 and 89 are not divisible by 7, and therefore they are prime, so those are the two prime numbers in that region.

In summary, we discussed the definition of prime numbers and listed the first few, we talked about the very important point, 1 is not a prime number, we talked about the other important point, 2 is the only even prime number, the lowest prime number, and the only even prime number, and we discussed the procedure for testing whether a number is prime

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