AS June 2023 Q5
5.
(i) Making your reasoning clear and using modulo arithmetic, show that\[214^6 \text{ is divisible by } 8\] (3)
(ii) The following 7-digit number has four unknown digits\[\boxed{a}\;5\;\boxed{b}\;8\;\boxed{a}\;\boxed{b}\;0\]Given that the number is divisible by 11
(a) determine the value of the digit \(a\). (2)
Given that the number is also divisible by 3
(b) determine the possible values of the digit \(b\). (3)
| Scheme | Marks | AO |
|---|---|---|
| \(214 \equiv 6 \pmod{8}\) | B1 | 1.1b |
| \((214)^2 \equiv 6^2 \pmod{8} \Rightarrow (214)^2 \equiv 4 \pmod{8}\) \((214)^6 \equiv 4^3 \pmod{8} \Rightarrow (214)^6 \equiv 64 \pmod{8}\) or \((214)^6 \equiv 6^6 \pmod{8} \equiv (6^3)^2 \pmod{8} \equiv (0)^2 \pmod{8}\) Or \((214)^3 \equiv 6^3 \pmod{8} \Rightarrow (214)^3 \equiv 0 \pmod{8}\) Or \(214 \equiv -2 \pmod{8} \Rightarrow (214)^6 \equiv (-2)^6 \pmod{8} \equiv 64 \pmod{8}\) | M1 | 2.1 |
| \((214)^6 \equiv 0 \pmod{8}\) therefore \(214^6\) is divisible by 8 | A1 | 2.4 |
| (3) |
Notes
B1: States \(214 \equiv 6 \pmod{8}\)
M1: Uses modulo arithmetic to find \((214)^6 \equiv a \pmod{8}\)
A1: Achieves \((214)^6 \equiv 0 \pmod{8}\) from correct modulo arithmetic and draws the conclusion that \(214^6\) is divisible by 8
| Scheme | Marks | AO |
|---|---|---|
| Uses \(a - 5 + b - 8 + a - b + 0 = 2a - 13 = 11n\) to find a possible value of \(a\). | M1 | 1.1b |
| \(a = 1\) | A1 | 2.2a |
| (2) |
Notes
M1: Uses the division rule for 11 to find a value for \(a\).
A1: \(a = 1\)
| Scheme | Marks | AO |
|---|---|---|
| \(\text{“}1\text{”} + 5 + b + 8 + \text{“}1\text{”} + b + 0 = 2b + 15 = 3n\) to find a possible value of \(b\). | M1 | 3.1a |
| Any two of \(b = 0,\ 3,\ 6\) or 9 | A1 | 1.1b |
| All four of \(b = 0,\ 3,\ 6\) and 9 | A1 | 2.2a |
| (3) | ||
| (8 marks) |
Notes
M1: Uses the division rule for 3 to find a value for \(b\).
A1: Any two correct values for \(b\).
A1: All four correct values for \(b\).