determine whether 117 is prime or composite by using the square root theorem

Question

hint- begin by taking its square root

Answer 1

Philip, I tried to post a comment but it would not post, so I'm posting it here.

You only have to divide by the prime numbers less than or equal to √n, not all the numbers.

Example: √731=27.037

You only have to divide by 23,19,17,13,11,7,5, and 3.

You don't have to divide by 26,24,22,20,18,16,14,12,10,8,6,4, and 2 because they are even numbers.

You don't have to divide by 25 (5*5), 21 (3*7), 15 (3*5), and 9 (3*3) because we are already checking for divisibility by

3,5, and 7. You only have to check for divisibility by 8 numbers instead of 26 numbers.

 Also, I know the directions were to use the square root theorem, but the student, knowing the rules for divisibility, should realize that 117 is divisible by 3 and by 9 because 1+1+7=9

Answer 2

A prime number is divisible only by itself and one.  So if n is a prime number, then its only positive factors are 1 and n: n = n*1.  So one way to prove that a number is prime is to divide it by all positive numbers up to the number itself.  If it divides evenly without a remainder for any number except itself and one, it is not prime.  For example:

 

4/1 = 4

4/2 = 2

4/3 = 1 remainder 1

4/4 = 1

 

So 4 is not a prime number because it's divisible by 2.  Let's try another:

 

5/1 = 5

5/2 = 2 remainder 1

5/3 = 1 remainder 2

5/4 = 1 remainder 1

5/5 = 1

 

This proves that 5 is a prime number.  We could do the same thing with 117, but we would have to divide 117 by one all the way up to 116, and that's a lot of work.  This where the square root theorem comes in. The theorem simply states that if a and b are factors of n (a*b=n), then one of them will be less than the square root of n.  This means that we only have to check up to +√n.  Now √117 ≅10.8, so to check if 117 is a prime number, we only have to check for divisibility up to 10.  Let's try it:

 

117/1 = 117

117/2 = 58 remainder 1

117/3 = 39 ----> we can stop right here. 117 = 39*3, so it is a composite number, not a prime number!