# Exam-Style Question on Number

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 182. This question is similar to one that appeared on a GCSE Higher paper (specimen) for 2017. The use of a calculator is allowed.

This expression can be used to generate a sequence of numbers.

$$n^2+n + 5$$

(a) Work out the first three terms of this sequence.

(b) What is the smallest value of $$n$$ that produces a term of the sequence that is not a prime number?

(c) Is it true that odd square numbers have exactly three factors? Explain and justify your answer.

(d) Seymour is thinking of a number.

• It is a common factor of 144 and 192.
• It is a common multiple of 6 and 8.
• It is less than 100.

Find the two possible numbers that Seymour could be thinking of.

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Here is a related challenge called the Prison Cell Problem.

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