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We must make use of our acquired mathematical knowledge. ![]() We cannot start from square one every time we explore a statement. (We will prove that this statement is true in the next section.) However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. We can do lots of calculation, such as \(3 7 = 10\) and \(5 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. If \(x\) and \(y\) are odd integers, then \(x y\) is an even integer. As an example, consider the conjecture that ![]() The best we can say is that our examples indicate the conjecture is true. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. ![]() Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false.
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