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

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #11 : How To Find Out If A Number Is Prime

What's the fourth smallest prime number?

Possible Answers:

\displaystyle 11

\displaystyle 4

\displaystyle 3

\displaystyle 5

\displaystyle 7

Correct answer:

\displaystyle 7

Explanation:

The order of the prime numbers start from \displaystyle 2, 3, 5, 7, 11, 13\displaystyle 1 is not prime as it's a unit. \displaystyle 4 is a composite number. So our fourth smallest prime number is \displaystyle 7.

Example Question #12 : How To Find Out If A Number Is Prime

Which is not prime?

Possible Answers:

\displaystyle 53

\displaystyle 39

\displaystyle 37

\displaystyle 31

\displaystyle 43

Correct answer:

\displaystyle 39

Explanation:

Since all the numbers are odd and don't end with a \displaystyle 5, let's check the basic divisbility rule. The divisibility rule for \displaystyle 3 is if the digits have a sum divisible by \displaystyle 3, then it is. 

\displaystyle 39=3+9=12

\displaystyle 31=3+1=4

\displaystyle 37=3+7=10

\displaystyle 43=4+3=7

\displaystyle 53=5+3=8

Based on this analysis, only \displaystyle 39 is divisible by \displaystyle 3 and therefore not prime and is our answer. 

Example Question #13 : How To Find Out If A Number Is Prime

Say \displaystyle p is a prime number. Which operation could possibly also lead to a prime number?

Possible Answers:

\displaystyle \textup{Subtraction}

\displaystyle \textup{Square root}

\displaystyle \textup{Exponents}

\displaystyle \textup{Multiplication}

\displaystyle \textup{Division}

Correct answer:

\displaystyle \textup{Subtraction}

Explanation:

Prime numbers are integers. So doing division and square roots will not generate integers. By doing multiplication and exponents, we involve more factors. The only possibility is subtraction. If \displaystyle p was \displaystyle 7 and we subtracted \displaystyle 2 we get \displaystyle 5 which is also prime. 

Example Question #14 : How To Find Out If A Number Is Prime

Say \displaystyle a is a number. \displaystyle a has, other than one and itself, only prime factors. \displaystyle a is not a perfect square. What is the smallest prime number \displaystyle a can be?

Possible Answers:

\displaystyle 8

\displaystyle 4

\displaystyle 2

\displaystyle 10

\displaystyle 6

Correct answer:

\displaystyle 6

Explanation:

Other than the number itself and one, we also need to have prime factors in that number. Since it's not a perfect square, we need to find the smallest possible prime numbers. That will be \displaystyle 2*3 or \displaystyle 6 which is our answer. 

Example Question #15 : How To Find Out If A Number Is Prime

Which of the following is not prime?

Possible Answers:

\displaystyle 983

\displaystyle 853

\displaystyle 691

\displaystyle 787

\displaystyle 687

Correct answer:

\displaystyle 687

Explanation:

Since all the numbers are odd and don't end with a \displaystyle 5, let's check the basic divisbility rule. The divisibility rule for \displaystyle 3 is if the digits have a sum divisible by \displaystyle 3, then it is. 

\displaystyle 687=6+8+7=21

\displaystyle 787=7+8+7=22

\displaystyle 853=8+5+3=16

\displaystyle 983=9+8+3=20

\displaystyle 691=6+9+1=16

Based on this analysis, only \displaystyle 687 is divisible by \displaystyle 3 and therefore not prime and is our answer. 

Example Question #16 : How To Find Out If A Number Is Prime

What's the largest prime number less than \displaystyle 100?

Possible Answers:

\displaystyle 91

\displaystyle 95

\displaystyle 89

\displaystyle 97

\displaystyle 93

Correct answer:

\displaystyle 97

Explanation:

Let's work backwards. All even numbers are not prime so we skip \displaystyle 98\displaystyle 99 is clearly divisible by \displaystyle 9\displaystyle 95 is definitely a composite number as it's divisible by \displaystyle 5\displaystyle 93 is divisible by \displaystyle 3 because of the divisibility rule \displaystyle (93=9+3=12).\displaystyle 91 is divisbile by \displaystyle 7. The divisibility rule for \displaystyle 7  is double the last digit and subtract from the rest \displaystyle (91=9-1*2=7). From the remaining answers, \displaystyle 97 is prime and is the largest prime number under \displaystyle 100 and is our answer. 

Example Question #17 : How To Find Out If A Number Is Prime

Which is prime?

Possible Answers:

\displaystyle 121

\displaystyle 132

\displaystyle 101

\displaystyle 99

\displaystyle 77

Correct answer:

\displaystyle 101

Explanation:

This will require us to know the divisibility rule of \displaystyle 11. The reason for this choice is that some of the numbers are palindromes like \displaystyle 11 so we eliminate \displaystyle 77, 99. For the three digit numbers, the divisibility rule for \displaystyle 11 is add the outside digits and if the sum matches the sum then it is divisible. Let's see.

\displaystyle 101=1+1\neq0

\displaystyle 121=1+1=2

\displaystyle 132=1+2=3

Based on this test, \displaystyle 101 is not divisible by \displaystyle 11 and is our answer.

Example Question #11 : Prime Numbers

If p\displaystyle p is a prime number, what could also be prime?

Possible Answers:

p^{2}\displaystyle p^{2}

3p\displaystyle 3p

p-2\displaystyle p-2

2p\displaystyle 2p

Correct answer:

p-2\displaystyle p-2

Explanation:

Plug in a prime number such as \displaystyle 7 and evaluate all the possible solutions. Note that the question asks which value COULD be prime, not which MUST BE prime. As soon as your number-picking yields a prime number, you have satisfied the "could be prime" standard and know that you have a correct answer.

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