Standard +0.8 This requires understanding modular arithmetic (n² ≡ 1 mod 10 for primes > 5) and constructing a multi-step proof, which is more conceptually demanding than routine AS-level algebra. However, the key insight (primes > 5 end in 1,3,7,9) is accessible, and squaring twice is straightforward once recognized, making it moderately challenging but not exceptional for a 5-mark proof question.
Question 7:
7 | Investigates last digit of n.
Allow M1 for investigation of 2k + 1 | AO3.1a | M1 | Last digit of n determines last digit
of n4
All even numbers divide by 2, so
are not prime
Any number ending in 5 is a
multiple of 5 so is not prime
Primes > 5 end in 1, 3, 7 or 9
If n ends in 1, 14 is 1 so n4 ends in a
1
If n ends in 3, 34 is 81 so n4 ends in
a 1
If n ends in 7, 74 is 2401 so n4 ends
in a 1
If n ends in 9, 94 is 6561 so n4 ends
in a 1
Statement proved by exhaustion
Deduces that only need to
investigate numbers ending in 1, 3,
7, 9
Condone inclusion of 5 at this
stage | AO2.2a | M1
Considers each in turn to show that
n4 will end in a 1 | AO1.1a | M1
Provides evidence that 14, 34, 74, 94
all end in a 1 | AO1.1b | A1
Constructs rigorous mathematical
argument to show the required
result
Only award if they have a
completely correct solution, which
is clear, easy to follow and contains
no slips.
Must include clear statement that
final digit of n determines final digit
of n4 | AO2.1 | R1
Total | 5
Q | Marking Instructions | AO | Marks | Typical Solution