Question 1141633: 3. Carmen is trying to determine if the following statement is always, sometimes, or never true.The square of a prime number is odd.She writes the following statements:3^2=9,5^2=25,7^2=49,11^2=121,13^2=169.Using her results, she concludes that the statement is always true.
(a) What kind of reasoning did Carmen use?
(b) Is her conclusion correct? If not, use a counterexample to prove why it is not.
Answer by jim_thompson5910(35256)   (Show Source):
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(a) She's using inductive reasoning. This is the kind of reasoning where you look at various examples and try to form a universal statement, rule or pattern based on the examples. As you'll see in part (b) below, this kind of thinking isn't always perfect.
(b) Her conclusion is not correct. The value 2 is prime, but squaring it leads to 2^2 = 2*2 = 4 which is not odd. Therefore the statement "The square of a prime number is odd" is not always true. It is only sometimes true. The reason why 2 doesn't work is because it is an even number (it is a multiple of 2). Squaring any even number leads to another even number. If she stated "The square of an odd prime number is odd", then her statement would always be true. Or she could say something like "For any prime p such that p > 2, the value p^2 is always odd". The prime p = 2 is the only even prime number. The use of this prime to disprove Carmen's conclusion is known as a counter-example as this example runs counter (or opposite) to what Carmen has concluded. It is the only possible counter-example for this problem.
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