What You Need to Know

A direct proof starts from known facts and uses logical steps to reach the conclusion

A counterexample disproves a general statement by providing one case where it fails

Algebraic proof uses variables to show a result holds for all numbers, not just specific ones

A conjecture is an unproven mathematical statement that appears to be true

Key Vocabulary

A rigorous logical argument that establishes a mathematical statement as definitely true

A single example that shows a general statement is false

A mathematical statement believed to be true but not yet formally proved

An integer is even if it equals 2k and odd if it equals 2k+1 for some integer k

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

To disprove the statement "all prime numbers are odd", the best approach is:

Prove it algebraically for all primes Find a counterexample (a prime that is even) Show it is true for the first 10 primes Use induction

Question 2

Which expression correctly represents an even integer?

2k + 1 k + 1 2k k²

Question 3

To prove algebraically that the sum of two odd numbers is always even, you would:

Check it for a few examples Use a calculator Draw a number line Let the odds be 2m+1 and 2n+1 and add them