What Is a Prime Number?
Default step content.
Those taking the GRE are expected to know what a prime number is. Okay, so what is it? A __prime number__ is a positive integer with only two positive factors: 1 and itself. Also, 1 is not considered to be prime.
Below are some examples illustrating this idea.
The number 11 is prime, because 11 is divisible by only 1 and itself.
The number 15 is not prime, because 15 is divisible by 3 and 5.
The number 23 is prime, because 23 is divisible by only 1 and itself.
Knowing this, which of the following is a list of all the prime numbers between 0 and 10?
Incorrect.
This list is incorrect for a couple of reasons. 1 and 9 are not primes. It might seem that 1 should be prime because it is divisible by only 1 and itself, but it only has one positive factor: 1. The number 9 is not prime because
$$\displaystyle 9 = 3 \cdot 3$$.
The fact that it is divisible by 3 means that it cannot be prime. Also, 5 is prime and it is excluded from the list. You know that 5 is prime because it is divisible only by 1 and itself.
Incorrect.
This list is incorrect. It is true that 3, 5, and 7 are all prime, but 2 is also prime.
The definition of a prime number is straightforward. But here is an interesting question: What is the largest even prime number?
Incorrect.
This has been known for over two thousand years!
Incorrect.
It's true that the numbers go on forever, but only one of them is even and prime.
To summarize this lesson:
[[summary]]
Correct!
The number 2 is prime because it is divisible only by 1 and itself. Also, any even number greater than 2 must be divisible by 2, so it cannot be prime. For example,
$$\displaystyle 300 = 2 \cdot 150$$;
$$\displaystyle 2,014 = 2 \cdot 1,007$$; and
$$\displaystyle 100,000 = 2 \cdot 50,000$$.
So 2 is the largest (and only) even number that is prime. It is an interesting fact that the primes go on forever, so there is no largest odd prime number. This has been known for over 2,000 years.
Correct!
Incorrect.
It is true that 2, 3, 5, 7 are all prime, yet 1 is not prime. It might seem like 1 should be prime because it is divisible only by 1 and itself, but according to convention it is not.
Default step content.
The numbers 2, 3, 5, and 7 are all of the prime numbers between 0 and 10. They are prime because they are divisible only by 1 and themselves. The numbers 1, 4, 6, 8, and 9 are not prime. The number 1 is not prime by definition, and 4, 6, and 8 are not prime because they are all divisible by 2. Finally, 9 is not prime because it is divisible by 3.
1, 2, 3, 7, 9
3, 5, 7
2, 3, 5, 7
1, 2, 3, 5, 7
Nobody knows.
There isn't one because the numbers go on forever.
The largest even prime number is 2.
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