# SAT Math : How to find out if a number is prime

## Example Questions

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### Example Question #1 : Prime Numbers

The sum of the first seven prime numbers divided by two is

24

29

58

61

33

29

Explanation:

The first seven primes are 2, 3, 5, 7, 11, 13, and 17. Don't forget about 2, the smallest prime number, and also the only even prime! Adding these seven numbers gives a sum of 58, and 58/2 = 29.

### Example Question #1 : Prime Numbers

How many integers between 2 and 20, even only, can be the sum of two different prime numbers?

Explanation:

There are 8 possible numbers; 4,6,8,10,12,14,16,18.

One is not a prime number, so only 8, 10, 12, 14, 16, and 18 can be the sum of two different prime numbers.

### Example Question #3 : Prime Numbers

Define a series of consecutive prime numbers to be a series of numbers, each prime, in which there are no other prime numbers between them.  These are not necessarily consecutive numbers themselves.  For example, the numbers 5,7 and 11 are consecutive prime numbers, although they are not consecutive numbers.

If  is the first number in a series of prime numbers, which of the following could not be the value of the last number in the series?

Explanation:

The primes, in order, are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ...

We create a few series:

-> series length 2: 2,3

-> series length 3: 3,5,7

-> series length 5: 5,7,11,13,17

-> series length 7: 7,11,13,17,19,23,29

etc.

We can then see that, of the answers, only 47 and 31 remain possibly correct answers.  Now we need to decide which of those two are impossible.

We could do another series, but the  series has 11 terms requiring us to go further and further up.  If we do this, we'll find that it terminates at 47, meaning that 31 must be the correct answer.

Another way, however, is to notice that 29 is the end of the  series.  Since 31 is the very next prime number, if we start on 11, the series that terminates in 31 would have to have length 7 as well.  Every series after  will thus end on a number larger than 31, meaning we will never finish on a 31.

### Example Question #1 : Prime Numbers

If  is a prime number, how many factors does  have?

Explanation:

The value of , or , is the product of  and , so it will be divisible by 1, p, p, and nothing else (we know that the p’s are not divisible because they are prime). Therefore p2 has exactly three factors.

(Alternatively, we can plug in any prime number for p and see how many factors p2 has. For example, if p is 3, then the factors of p2, or 9, are 1, 3, and 9.)

### Example Question #1 : Prime Numbers

The sum of four consecutive integers is 210.  Which one of these four integers is prime?

57

51

53

47

49

53

Explanation:

Let x represent the smallest of the four numbers.

Then we can set up the following equation:

Therefore the four numbers are 51, 52, 53, 54.  The only prime in this list is 53.

### Example Question #6 : Prime Numbers

Which number is prime?

Explanation:

A prime number is a number with factors of one and itself.

Let's try to find the factors.

It may not be easy to see  as a composite number, but if you know the divisibility rule for  which is double the last digit and subtract from the rest , you will see  is not prime.

### Example Question #1 : Prime Numbers

Which is the smallest prime number?

Explanation:

The smallest prime number is actually  is not a prime nor a composite number. It is a unit.

### Example Question #8 : Prime Numbers

Which of the following is a prime number?

Explanation:

A prime number is a number with factors of one and itself.

Let's try to find the factors.

It may not be easy to see  as a composite number, but if you know the divisibility rule for  which is double the last digit and subtract from the rest, you will see  is not prime. The divisibility rule for  is add the outside digits and if the sum matches the sum then it is divisible . The divisibility rule for  is if the digits have a sum divisible by , then it is . All even numbers are composite numbers with the exception of . So with these analyses, answer is

### Example Question #9 : Prime Numbers

Which of the following is NOT prime?

Explanation:

Since all of the digits don't add to a sum of , and we dont see any numbers een or end in , let's try the divisibility rule for  which is double the last digit and subtract from the rest.

Only  is divisible by  and is not prime and therefore our answer.

### Example Question #10 : Prime Numbers

What are the first three prime numbers?

Explanation:

The smallest prime number is actually  is not a prime nor a composite number. It is a unit. This will eliminate the choices with a  in them. The next prime numbers are . Our answer is then  is a perfect square and has more than two factors .

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