# Exam-Style Questions.

## Problems adapted from questions set for previous Mathematics exams.

### 1.

GCSE Higher

Use algebra to prove that $$0.3\dot1\dot8 \times 0.\dot8$$ is equal to $$\frac{28}{99}$$.

### 2.

GCSE Higher

m and n are positive whole numbers with m > n

m2 – n2 = (m + n)(m – n)

If m2 – n2 is a prime number prove that m and n are consecutive numbers.

### 3.

GCSE Higher

$$\frac{p-1}{q-1}-\frac pq$$

Using your answer to part (a), prove that if $$p$$ and $$q$$ are positive integers and $$p \lt q$$, then

$$\frac{p-1}{q-1}-\frac pq\lt 0$$

### 4.

GCSE Higher

(a) Prove that the product of two consecutive whole numbers is always even.

(b) Prove, by giving a counter example, that the sum of four consecutive integers is not always divisible by 4.

### 5.

IB Analysis and Approaches

Consider two consecutive positive even numbers, $$2n$$ and $$2n + 2$$.

Show that the difference of their squares is equal to twice their sum.

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