Digital equation
Problem
Let $k$ be an integer satisfying $0\le k \le 9\,$.
Show that $0\le 10k-k^2\le 25$.
Show also that $k(k-1)(k+1)$ is divisible by $3\,$.
When considering $k(k-1)(k+1)$ it might be helpful to rearrange the factors.
It might also help to consider the cases $k=3b$, $k=3b+1$ and $k=3b+2$.
Why does this cover all possible values of $k$?
For each $3$-digit number $N$, where $N\ge100$, let $S$ be the sum of the hundreds digit, the square of the tens digit and the cube of the units digit. Find the numbers $N$ such that $S=N$.
[Hint: write $N=100a+10b+c\,$ where $a\,$, $b\,$ and $c$ are the digits of $N\,$.]
What values do you know that $a$, $b$, $c$ lie between? Since $N$ is a three digit number we know that $a \neq 0$.
Can you see how the first parts of the question might be related to this part?
STEP Mathematics 1, 2003, Q7. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.
Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.
Getting Started
- If $k$ is an integer between 0 and 9, what possible values can $10k-k^2$ take? What are the minimum and maximum values?
- What do you know about the factors $k$, $k-1$ and $k+1$? What does this mean about the product of these three factors?
- What is $S$ in terms of $a$, $b$ and $c$? Make sure you read the question carefully!
- Can you use $S=N$ to write an equation involving $a$, $b$ and $c$?
- What values can $a$, $b$, $c$ take? What does the fact that $N$ is a 3-digit number tell us about $a$?
- Can you link the first parts of the question to the last part?
- Can you put bounds on the values of the parts of the equation involving $a$, $b$, and $c$?
- Can you rule out any values of $a$, $b$ or $c$?
Student Solutions
Let $k$ be an integer satisfying $0\le k \le 9\,$. Show that $0\le 10k-k^2\le 25$.
Mal from Bedford School in the UK and Mahdi from Mahatma Gandhi International School in India tried all the possible values of $k.$ Here is Mal's work:
Nikita from Jersey College for Girls, Mohamed S from LAE Tottenham in the UK and Sanika from PSBBMS, OMR in India used a graph. The sketch is from Nikita's work. This is how Sanika found the roots and the maximum value:
The roots of the equation are 0 and 10.
Since the squared term has a negative coefficient the parabola will face downwards.
The highest y-coordinate will therefore be in the first quadrant with an x value right between the two x intercepts. (As quadratic equations have symmetric graphs). This implies that the x coordinate is (0+10)$\div$2 = 5.
Substituting this value into the function $f(x)$ will give us the y coordinate; this will be the highest output of the function.
$\therefore 25\geq k(10-k)$
If the condition that $0\leq k\leq9$ is considered, then it can be concluded that the lowest y-coordinate is obtained when the x-coordinate is $0$ $\therefore 0\leq k(10-k)$
Nikita found the maximum value by completing the square:
$y=-(k-5)^2+25$
so max point is at $(5,25)$
Mohamed S found the maximum value using differentiation:
$f'(k) = 10-2k$
To find maximum:
$10-2k=0$
$10=2k$
$k=5$
$10\times5 - 5^2 = 25$
$25$ is the maximum
Wiktor from LAE Tottenham and David from the UK proved the inequality algebraically. This is David's work:
Sanika also proved the inequality using the arithmetic mean - geometric mean inequality:
AM-GM inequality: $\dfrac{a+b}2 \geq \sqrt{ab}$
Applying AM-GM to the numbers $k$ and $(10-k)$ we get the following: $$\frac{k+(10-k)}2 \geq \sqrt{k(10-k)} \Rightarrow 25\geq k(10-k)$$The expression will always give a positive integer as output if $0\lt k \leq 9\,.$ So the lowest value can be obtained when $k=0\, .$
Show also that $k(k-1)(k+1)$ is divisible by $3\,$.
Mal checked this for the integers 1 to 9:
David, Wiktor, Mohamed S, Mahdi and Nikita wrote this simple proof (this is Wiktor's work):
Nikita and Sanika considered different cases. This is Nikita's work:
For each $3$-digit number $N$, where $N\ge100$, let $S$ be the sum of the hundreds digit, the square of the tens digit and the cube of the units digit. Find the numbers $N$ such that $S=N$.
Nikita, David, Mal, Wiktor, Mohamed S, Sanika and Mahdi found an equation linking $a,b$ and $c\,.$ This is Nikita's work:
Nikita then found the possible values of $c^3-c$ and went through the values of $a$ to find solutions.
Wiktor and David used part 1 to find out more about $b$. This is Wiktor's work:
Mohamed wrote:
This means that a multiple of $99$ minus a number between $0$ and $25$ must be equal to $c(c+1)(c-1).$ $99a$ must be close to $c(c+1)(c-1)$ but smaller.
Mahdi, Mohamed, Wiktor and Mal used this idea to go through the possible values of $c$. This is Mahdi's work:
Using all the information, Sanika made a table showing the possible values of $a, b$ and $c$:
|
All possible values of: |
All possible values of: |
All possible values of: |
|||
|
$99a$ |
$a$ |
$b(10-b)$ |
$b$ |
$c(c-1)(c+1)$ |
$c$ |
|
99 |
1 |
0 |
0 or 10 |
120 |
5 |
|
198 |
2 |
9 |
1 or 9 |
210 |
6 |
|
297 |
3 |
21 |
3 or7 |
336 |
7 |
|
396 |
4 |
24 |
4 or6 |
504 |
8 |
|
495 |
5 |
|
|
720 |
9 |
|
594 |
6 |
|
|
|
|
|
693 |
7 |
|
|
|
|
After a couple of combinations, 4 numbers can be derived:
175
135
518
598
Teachers' Resources
Why do this problem?
This problem explores some equations involving integers and number theory. It also explores how number theory facts can be used to find all integer solutions to a problem. There are several ways of approaching this problem, including checking manually all the possibilities.
STEP papers are examinations taken at the end of Year 13 and are used by Cambridge, Warwick and a number of other universities in the UK as part of their offer (or part of a range of offers) to Maths applicants. The questions are based on A Level Maths and Further Maths, but differ in style to A level questions. This question can be attempted with no A level knowledge at all and so is a good introduction to university entrance tests for Year 12 (and Year 11) students.
Possible approach
Students could be asked to find more than one way of showing that $0 \le 10k-k^2 \le 25$. When considering $k(k-1)(k+1)$ they could consider a few values and then try to show that it is always divisible by 3 (though the question only asks you to show it for $0\le k \le 9$).
For the second part of the question, students should start by trying to find an equation involving $a$, $b$ and $c$. They should gather up terms and then see if they can connect their equation to the first part of the question. It might be easier to isolate the $c$ terms on one side of the equation.
Key questions
- If $k$ is an integer between 0 and 9, what possible values can $10k-k^2$ take? What are the minimum and maximum values?
- What do you know about the factors $k$, $k-1$ and $k+1$? What does this mean about the product of these three factors?
- What is $S$ in terms of $a$, $b$ and $c$? Make sure you read the question carefully!
- Can you use $S=N$ to write an equation involving $a$, $b$ and $c$?
- What values can $a$, $b$, $c$ take? What does the fact that $N$ is a 3-digit number tell us about $a$?
- Can you link the first parts of the question to the last part?
- Can you put bounds on the values of the parts of the equation involving $a$, $b$, and $c$?
- Can you rule out any values of $a$, $b$ or $c$?
Possible support
Students could try the problem Rational Integer where they are encourage to work through possible integer values systematically. They might want to restrict themselves to positive integers to fit in with this problem more closely. Students could also consider the first question from Common Divisor.
Encourage students to consider the key questions above. A systematic approach, trying out values, is not cheating!
Possible extension
Students who have enjoyed this problem might like to explore the assignments from the STEP Support Programme.
We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.