Primes play a central part in integer questions. Overconfidence is dangerous here: while almost everybody can recite the definition of a prime number at the drop of a hat, the field is actually rife with misconceptions. We are here to make sure you know everything there is to know about primes.
To understand primes, let’s first take a look at the definition of a prime:
“A prime number is a positive integer with exactly two distinct positive factors: 1 and itself”.
5 is a prime number because it has only two distinct positive factors: 5 and 1.
12 is not prime, because it has more than two factors: 1, 2, 3, 4, 6, and 12 are all factors of 12.
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If x is a prime number, which of the following CANNOT be a value of x?
(A) Correct By definition, a prime must be a positive integer, so x cannot be 0. (B) Incorrect The only positive factors of 3 are 1 and 3, and is therefore prime. Note that the question asks which of the following CANNOT be a value of x. (C) Incorrect The only positive factors of 7 are 1 and 7, and is therefore prime. Note that the question asks which of the following CANNOT be a value of x. (D) Incorrect The only positive factors of 11 are 1 and 11, and is therefore prime. Note that the question asks which of the following CANNOT be a value of x. |
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If x is a prime number, then which of the following CANNOT be the value of x?
1 is often mistakenly considered prime, because it is divisible by 1 and itself, but those are not two distinct factors – they’re the same factor. Therefore, by definition, 1 is not prime. |
Okay, so if negative numbers and zero are not prime, and 1 is not prime either, Then the smallest prime integer must be?
2 has only two positive factors, i.e. 1 and itself (2). That makes 2 the smallest prime number.
Memorizing the list of primes up to 50 is helpful for quickly working out integer questions. The primes up to 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Note something interesting about the above list: most of the primes are odd. In fact, 2 is the only even prime on that list. Coincidence?
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Could there be another even prime other than 2? Any even number is divisible by 2. So every positive even integer (other than two) will have at least 3 positive factors: 1, itself, and 2, and will therefore not be prime. |
Remember this about 2:
- 2 is the smallest prime.
- 2 is the only even prime.
A composite number is an integer greater than 1 that is not a prime number. The first five composite numbers are 4, 6, 8, 9, and 10.
To sum up our lesson:
A prime number is a positive integer with exactly two distinct positive factors: 1 and itself.
Remember the following facts about primes:
- 1 is not considered prime.
- 2 is the smallest prime.
- 2 is the only even prime.
A composite number is an integer greater than 1 that is not a prime number. The first five composite numbers are 4, 6, 8, 9, and 10.