Answer to Question #306250 in Discrete Mathematics for Frank

Question

Find out if the following numbers are prime numbers, show your work using prime factorization. You may use code to verify your answer but do not put it up as your solution.:

(a) - (3 points) 773

(b) - (3 points) 733

(c) - (3 points) 377

Answer 1

We know for any non-negative integer n; prime numbers greater than 3, always satisfy the 6n + 1 or 6n – 1 condition.

 

Answer (a)

For the number, 773, let us write it in the form, 6n-1,

6n – 1 = 773

6n = 773 + 1

6n = 774

n = 129

Hence, 773 can be written in the form 6 n – 1, for n = 129

Therefore, 773 is a prime number.

Furthermore, 773 has only 1 and 773 as its two factors.

 

Answer (b)

For the number, 733, let us write it in the form, 6n – 1,

6n – 1 = 733

6n = 733 + 1

6n = 734

n = 367/3 (not a positive integer)

 

 

Let us try to write 733, in the form 6n + 1

6n + 1 = 733

6n = 733 - 1

6n = 732

n = 122  

Hence for n = 122, the number 733 can be written in the form 6 n + 1, so

733 is a prime number.

Furthermore, 733 has only 1 and 733 as its two factors.

 

Answer (c)

For the number, 377, let us write it in the form, 6n – 1,

6n – 1 = 377

6n = 377 + 1

6n = 378

n = 63

Hence for n = 68, the number 377 can be written in the form 6 n – 1, so

377 is a prime number.

Furthermore, 377 has only 1 and 377 as its two factors.