Exam-Style Questions.

1.	GCSE Higher

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

Worked Solution

---

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.

Worked Solution

---

3.	GCSE Higher

Express as a single fraction and simplify your answer.

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

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

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

Worked Solution

---

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.

Worked Solution

---

5.	GCSE Higher

The number \(T\) can be expressed as \(T = 4k + 3\) where \(k\) is a positive integer.

(a) Show that \(T^2\) is always an odd number.

\(T\) and \(U\) are consecutive odd numbers where \(U > T\).

(b) Write down an expression for \(U\), in terms of \(k\).

(c) Show that \(U^2 - T^2\) is always a multiple of 16.

Worked Solution

---

6.	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.

Worked Solution

---