EasyMathematicsHigh School (9–10)GeneralNumber Systems › Rational and Irrational NumbersNumericalBoard Exam

Kabir works out n² - n + 11 for n = 1, 2, 3, … and finds 11, 13, 17, 23, 31 and so on, all primes…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Kabir works out $n^2 - n + 11$ for $n = 1, 2, 3, \ldots$ and finds $11$ , $13$ , $17$ , $23$ , $31$ and so on, all primes. Find the smallest $n$ for which $n^2 - n + 11$ is not a prime, and the value it gives. What does this show about checking examples?

Need a reminder? The key ideaShow

In a proof by contradiction we suppose that the statement is false, reason correctly, and reach something impossible. Then the statement must be true. Examples alone are not a proof, but one counterexample shows that a statement is false.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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