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2 votes
Give a counterexample

for the following
conjecture: "an odd number plus another odd
number will always be divisible by 4".
13.

User Benja
by
7.8k points

1 Answer

6 votes

Answer:

5 and 25

Explanation:

A single counterexample to any conjecture that states that something will always be true immediately disproves the statement.

Conjecture

Conjecture is a statement or conclusion that is offered with little to no proof. The conclusion given in the question has no mathematical proof to support the idea; thus, it is conjecture. Conjecture is often easy to disprove.

Counterexamples are examples that follow the conditions of a statement (or conjecture) but do not meet the same conclusion. The conditions of the conjecture given are "an odd number plus another odd number". So, a counterexample must also meet these conditions. The conclusion of the conjecture is that this sum " will always be divisible by 4". Thus, the counterexample should not be divisible by 4.

Counterexample

At first, the conjecture may seem to be true because of pairs like 13+15 or 21+23. However, because the conjecture includes the word "always", it only takes one counterexample to completely disprove the statement.

I know that 30 is not divisible by 4, so I can find 2 odd numbers that sum to 30. I pair I chose was 5+25, but there are other possibilities like 13+17 or 1+29. No matter which you choose, it's clear that the conjecture is not true.

User Claude Hasler
by
7.7k points