Mathematics · Oct / Nov 2023 · Paper 21 · Question 1

YMA01

As printed in the paper

The question as text

1. Given that a, b and c are integers greater than 0 such that • c = 3a + 1 • a + b + c = 15 prove, by exhaustion, that the product abc is always a multiple of 4 You may use the table below to illustrate your answer. (3) You may not need to use all rows of this table. a b c abc

Read automatically from the paper; the printed page above is exact.

Show the answer and mark scheme

abc

One correct set a, b & c

Two correct rows

All values of abc are multiples of 4

Fully correct + statement

Main method: Numerical solution finding all known solutions

B1: Any correct set of values for a, b and c. You can ignore any attempt at the product abc

M1: Any two fully correct rows including the correct calculations for the products.

Condone extra/ incorrect rows

A1: Fully correct. The solution must have

the three correct rows with correct calculations for abc

any extra (or incorrect) rows must be deleted or somehow excluded

a minimal statement showing that they are aware that 40, 84 and 60 are multiples of 4. E.g they are all

divisible by 4, use of a tick Π, prove, QED, calculations such as 40

4 =

. They can be divided by 4 (on

its own) is insufficient without further clarification such as ''to give whole numbers''

no incorrect calculations. E.g if their attempt to prove that 60 being a multiple of 4 includes 60

then it is A0

Alt 1: Via deduction

B1: Either

Solves a + b + c = 15 with c = 3a + 1 to conclude that 4a + b = 14 o.e. and states that b is even

Uses

+ to state that if a is odd then c is even and if a is even then c is odd

M1: Attempts to generalise and offers one of the two possibilities for the product abc.

Either

states then when a is odd, b is even and c is even, therefore abc = odd

even even

states then when a is even, b is even and c is odd, therefore abc = even even odd

A1: Fully correct. The solution must have

the two possible ''calculations'' for abc.

no incorrect ''calculations''. E.g. if they state that even even

is a multiple of 4 because

is A0. It must be along the lines of 2

with different variables.

You may see a hybrid solution that can be marked along similar lines. Mark positively.

If you see a tabular solution and one via deduction, award marks for the better one.

Please use review if you are uncertain.

2 ,

Practise this paper →

Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.