Math 130

Fall 03

 

Number Theory: Definitions and Conjectures

 

 

1. Definitions

 

List each type of number.  Use ellipses (…).

 

Natural Numbers:

 

Integers:

 

Positive Integers:

 

Negative Integers:

 

Non-Negative Integers:

 

Finish each definition. 

 

            1.  If  is an integer and  is a natural number, then  is a multiple of  if

 

 

 

            2.  If  and  are natural numbers, then  is divisible by  if

 

 

 

            3.  If  and  are natural numbers, then  is not divisible by   if

 

 

            4.  If  and  are natural numbers, then  is a factor of   if

 

 

            5.  If  is an integer then  is even if

 

 

            6.  If  is an integer then  is odd if

 

 

7.  If  is a natural number, then  is prime if

 

 

8. If  is a natural number, then  is composite if

2.  Conjectures to Evaluate:

 

            Below are several conjectures inspired by our last class.  Decide whether each conjecture is true or false.  Give counter-examples to false conjectures and see whether you can modify them to make them true.  Try to prove why true conjectures are true in all cases.  Use your definitions from part 1.

            Feel free to consult section 3: “General notes on Proving/Disproving Conjectures.”

 

Assume that  is a natural number.

 

1.  0 is even.

 

 

 

 

2.  0 can be even or odd.

 

 

 

 

3.  If the last digit of  is 4, then  is divisible by 4.

 

 

 

 

 

4.  If the last digit of  is 3, 6, or 9, then  is divisible by 3.

 

 

 

 

 

5. If the last digit of  is 0 or 5, then  is divisible by 5.

 

 

 

 

 

6.  If the last digit of  is even, then  is even.

 

 

 

 

 

7.  If  is even, then its last digit is even.

8.  If  is divisible by 2 and 3, then  is divisible by 6.

 

 

 

 

 

9.  If  is divisible by 6, then  is divisible by 2 and 3.

 

 

 

 

 

10.  If  is divisible by 2 and 4, then  is divisible by 8.

 

 

 

 

 

11.  If  is divisible by 8, then  is divisible by 2 and 4.

 

 

 

 

 

12. If  is divisible by 9, then the sum of its digits is 9.

 

 

 

 

 

13. If  is a multiple of 4, then twice the first digit of  plus the second digit of  is also a multiple of 4.

 

 

 

 

 

14. If twice the first digit of  plus the second digit of  is a multiple of 4

then  is a multiple of 4.

 

 

 

 

 

 

15.  If  is a multiple of 11, then all its digits are the same.

 

16.  If  is a three-digit multiple of 11, then the second digit of  is one more then the third digit of .

 

 

 

 

 

17.  If  is a two-digit multiple of 12, then four times the sum of the digits of  is equal to .

 

 

 

 

18.  If  is divisible by 12, then  is divisible by 2, 3, 4, and 6.

 

 

 

 

 

 

19.  If  is divisible by 3 and 4, then  is divisible by 12.

 

 

 

 

 

 

20.  If  is divisible by 2 and 6, then  is divisible by 12.

 

 

 

 

Summarize what that last digit of  says about the  ’s divisibility by various natural numbers: