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.
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