Discrete Mathematics: The Math of Computer Science

Problem 1 

The number below is rational. Thus, write it as a ratio of two integers. 

Solution 

Multiply equation (1) both sides by 10 to get 

Multiply equation (2) both sides by 10 to get 

Multiply equation (3) both sides by 10 to get 

Multiply equation (4) both side by 10 to get 

Multiply equation (5) both side by 10 to get 

Multiply equation (6) both sides by 10 to get 

Subtracting equation (3) from equation (7) we get 

Problem 2 

Prove or disprove the following statement. For all integers a, b, and c, if a | bc then a | b or a | c . 

Solution 

Suppose first that

Then there is an integer such that 

Then

Hence, if the

We prove the converse 

Suppose

Then there is an integer such that 

Then

Hence, is then

Therefore, the statement is true.