CAIE Further Paper 1 2023 June — Question 1 6 marks

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2023
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve divisibility
DifficultyStandard +0.3 This is a standard proof by induction for divisibility, requiring verification of the base case (n=1) and an inductive step showing 5^(3(k+1)) + 32^(k+1) - 33 is divisible by 31 given the assumption for n=k. The algebraic manipulation is straightforward using factoring and the fact that 5^3 = 125 ≡ 1 (mod 31) and 32 ≡ 1 (mod 31), making this slightly easier than average but still requiring proper induction structure.
Spec4.01a Mathematical induction: construct proofs
1 Prove by mathematical induction that, for all positive integers \(n , 5 ^ { 3 n } + 32 ^ { n } - 33\) is divisible by 31 .
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(5^3 + 32 - 33 = 124\) is divisible by 31.B1 Checks base case.
Assume that \(5^{3k} + 32^k - 33\) is divisible by 31 for some positive integer \(k\).B1 States inductive hypothesis.
\(5^{3k+3} + 32^{k+1} - 33 = (124+1)5^{3k} + (31+1)32^k - 33\)M1 A1 Separates \(5^{3k} + 32^k - 33\) or considers difference.
is divisible by 31 because \(124 \times 5^{3k} + 31 \times 32^k\) is divisible by 31.A1
Hence, by induction, true for every positive integer \(n\).A1
Total6 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $5^3 + 32 - 33 = 124$ is divisible by 31. | B1 | Checks base case. |
| Assume that $5^{3k} + 32^k - 33$ is divisible by 31 for some positive integer $k$. | B1 | States inductive hypothesis. |
| $5^{3k+3} + 32^{k+1} - 33 = (124+1)5^{3k} + (31+1)32^k - 33$ | M1 A1 | Separates $5^{3k} + 32^k - 33$ or considers difference. |
| is divisible by 31 because $124 \times 5^{3k} + 31 \times 32^k$ is divisible by 31. | A1 | |
| Hence, by induction, true for every positive integer $n$. | A1 | |
| **Total** | **6** | |

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1 Prove by mathematical induction that, for all positive integers $n , 5 ^ { 3 n } + 32 ^ { n } - 33$ is divisible by 31 .\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q1 [6]}}
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Q1 6 Q2 8 Q3 9 Q4 14 Q5 10 Q6 15 Q7 13