Year: 2020
Paper: 2
Question Number: 5
Course: UFM Additional Further Pure
Section: Number Theory
No solution available for this problem.
There were just over 800 entries for this paper, and good solutions were seen to all of the questions. Candidates should be aware of the need to provide clear explanations of their reasoning throughout the paper (and particularly in questions where the result to be shown is given in the question). Short explanatory comments at key points in solutions can be very helpful in this regard, as can clearly drawn diagrams of the situation described in the question. The paper included a few questions where a statement of the form "A if and only if B" needed to be proven – candidates should be aware of the meaning of such statements and make sure that both directions of the implication are covered clearly. In general, candidates who performed better on the questions in this paper recognised the relationships between the different parts of each question and were able to adapt methods used in earlier parts when working on the later sections of the question.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
If $x$ is a positive integer, the value of the function $\mathrm{d}(x)$ is the sum of the digits of $x$ in base 10. For example, $\mathrm{d}(249) = 2 + 4 + 9 = 15$.
An $n$-digit positive integer $x$ is written in the form $\displaystyle\sum_{r=0}^{n-1} a_r \times 10^r$, where $0 \leqslant a_r \leqslant 9$ for all $0 \leqslant r \leqslant n-1$ and $a_{n-1} > 0$.
\begin{questionparts}
\item Prove that $x - \mathrm{d}(x)$ is non-negative and divisible by $9$.
\item Prove that $x - 44\mathrm{d}(x)$ is a multiple of $9$ if and only if $x$ is a multiple of $9$.
Suppose that $x = 44\mathrm{d}(x)$. Show that if $x$ has $n$ digits, then $x \leqslant 396n$ and $x \geqslant 10^{n-1}$, and hence that $n \leqslant 4$.
Find a value of $x$ for which $x = 44\mathrm{d}(x)$. Show that there are no further values of $x$ satisfying this equation.
\item Find a value of $x$ for which $x = 107\mathrm{d}\left(\mathrm{d}(x)\right)$. Show that there are no further values of $x$ satisfying this equation.
\end{questionparts}
This was a popular question and many of the solutions made good progress on the early parts of the question. The majority of candidates gained full marks for part (i), but some candidates did not mention that x − d(x) ≥ 0. There was a wide range of marks achieved on part (ii). The proof that x − 44d(x) is a multiple of 9 if and only if x is a multiple of 9 was completed well by those who managed to prove the result, but the majority of other attempts seen did not score any marks. In a small number of cases only the "if" direction was proved. Those who were unable to prove the first result in this part were often able to continue and find the required bounds on x however. Candidates who had completed both of these parts generally managed to find the correct answer x = 792, but did not necessarily fully justify that it was the only one. Most candidates scored low marks on part (iii). It was very common to see an insufficient proof that 9|x. Without guidance from the question as to how to find bounds on x, students produced a wide range of approaches; better bounds were needed if the student only used 107|x, but the simple bound d(d(x)) ≤ d(x) together with divisibility by 963 was sufficient.