Frame the converse for the proposition. Then, determine if each of the two statements is true or not. Justify the true statements
Proposition P: Whenever \(n\) is a perfect square, the total number of its factors is odd.
Converse Q : If then
Justification:
Proportion P: Let \(n\) be a perfect square.
For example, \(n=25=\) .
The factors of 25 are:
There are factors, which is an number.
In general, the factors of a perfect square occur in pairs, except for its square root, which pairs with itself.
Therefore, a perfect square has an number of factors.
Hence, Proposition P is .
Converse Q: Suppose \(n\) has an odd number of factors.
For example, \(n=\)25.
Its factors are , giving factors, which is .
Also, \(n=25=\) a perfect square.
In general, if a positive integer is not a perfect square, its factors occur in distinct pairs. Hence, it has an even number of factors.
Therefore, a positive integer with an number of factors must be a perfect square.
Hence, Converse Q is also .
Both Proposition P and Converse Q are .