Edexcel IGCSE Math A Higher 2.2 He Algebra to Support and Construct Proofs Questions

Practise representing consecutive or restricted integers algebraically and rearranging results to prove divisibility or parity.

Exam points

represent consecutive integers, even numbers or multiples using n-based expressions
expand and simplify both sides until the claimed difference or equality is visible
factor the result to exhibit an integer multiple or the form 2k+1 for oddness

[Maximum number: 3]

Here are the first four terms of a sequence of fractions.

\frac{1}{1} \quad \frac{2}{3} \quad \frac{3}{5} \quad \frac{4}{7}

The numerators of the fractions form the sequence of whole numbers $1234 \ldots$ The denominators of the fractions form the sequence of odd numbers $1 \quad 3 \quad 5 \quad 7 \ldots$

Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1

Let an odd number be 2n-1.
$(2n-1)^2=4n^2-4n+1$ $=4(n^2-n)+1$
Therefore when the square is divided by 4, the remainder is 1.

M1 for using $(2n-1)^2$ or $(2n+1)^2$ .
M1 for writing the expression as a multiple of 4 plus 1.
A1 for a correct conclusion.

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Edexcel IGCSE Math A Higher 2.2 He Algebra to Support and Construct Proofs Questions question 1