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The sum of digits in a two digit number formed by the two digits from $1$ to $9$ is $8$. If $9$ is added to the number then both the digits become equal. Find the number.

My attempt:

Let the two digit number be $10x+y$ where, $x$ is a digit at tens place and $y$ is the digit at unit's place. According to question:

$$x+y=8$$

I could not figure out the other condition. Please help. Thanks in advance.

Rodrigo de Azevedo
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pi-π
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    $\overline{xy} = 10x+y$ is the number in question. $x+y = 8$, and $\overline{xy}+9 = 10x+y+9 = 10(x+1)+(y-1) = \overline{(x+1)(y-1)}$, and $x+1=y-1$. – 2012ssohn Jan 04 '17 at 03:12
  • @2012ssohn, what is $\overline {xy}$? How did you get to this? – pi-π Jan 04 '17 at 03:16
  • $\overline{xy}$ is a way of writing that $x$ and $y$ are digits - $xy$ could lead to some confusion as it can mean either $10x+y$ or $x \cdot y$. – 2012ssohn Jan 04 '17 at 03:18
  • @2012ssohn, It means we can either write $10x+y$ or $\overline {xy}$? – pi-π Jan 04 '17 at 03:21
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    Yes - $\overline{xy}$ is just a simpler notation of writing the same thing. If this were a five-digit number, for example, I would prefer writing $\overline{abcde}$ instead of $10000a+1000b+100c+10d+e$. – 2012ssohn Jan 04 '17 at 03:22
  • @2012ssohn, could you please explain me the second condition? I could not get it clearly. – pi-π Jan 04 '17 at 03:26
  • "If $9$ is added to the number" - $9$ refers to the original number, in this case, $\overline{xy}$. This effectively transforms the number $\overline{xy}$ to $\overline{zz}$, which has two equal digits. In other words, $\overline{xy}+9=\overline{zz}$. – 2012ssohn Jan 04 '17 at 03:27

6 Answers6

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First note that $y \ne 0$ since otherwise we would have $x + y = x + 0 = 8$, and so $10x + y = 80$, but $80$ doesn't satisfy the second condition.

Therefore we must have $1 \le y \le 9$. This means that when we add $9$ to $10x + y$, the tens digit must increase by $1$ and the ones digit decreases by $1$. So then $10x + y + 9 = 10(x+1) + (y-1)$. Since the digits are equal, we have $x+1 = y-1$. Now you just have a system of two equations in two variables:

\begin{align*} x+y &= 8\\ x+1 &= y-1 \end{align*}

  • How is $10x+y=80$? – pi-π Jan 04 '17 at 03:18
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    @user354073, I was explaining why $y \ne 0$, and part of that explanation is $10x + y = 80$. $\color{red}{\textbf{If}}$ $y = 0$, then the condition $x+y = 8$ tells us $x = 8$. Therefore $10x + y = 10(8) + 0 = 80$. But $80 + 9 = 89$ doesn't satisfy the requirement that both digits are equal when $9$ is added to the original number. Therefore $y$ can't be zero. And I wanted to show $y \ne 0$ in order for my second paragraph in the answer to work. –  Jan 04 '17 at 03:24
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    @tilper Nice answer. +1 – user401699 Jan 04 '17 at 03:32
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Let, number is of the form $10a+b$, then according to question:

$a+b=8$ and digits of $10a+b+9$ are equal.

Notice that adding $9$ to the give number will increase its tens digit by $1$ and decrease its unit digit by $1$. So,

$a+1=b-1\implies a+2=b$.

Hence, we have $a+2+a=8\implies a=3\implies b=5 \implies 10a+b=35$

Vidyanshu Mishra
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    yeah, @tilper, though it will still decrease the unit digit as 10-9, (Sort of :) :)) – Vidyanshu Mishra Jan 04 '17 at 03:35
  • @THE LONE WOLF, Why is the digits $a+1$ and $b-1$ when $9$ is added? Why the digit at tens place increases and that at ones place decreases? – pi-π Jan 04 '17 at 04:00
  • @user354073, note the word notice that. I just checked a pattern which was clear to see. As 23+9=32, 67+9=76 etc – Vidyanshu Mishra Jan 04 '17 at 04:27
  • @user354073 The question states the digits are in the range 1-9 (i.e. not zero). Adding 9 to any number with a nonzero units digit will cause a carry. So the tens digit becomes a+1 while the units digit becomes b+9-10=b-1 – Peter Green Jan 04 '17 at 14:46
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The number is $35$, since $$x+y=8$$ and if $9$ is added to any number the ones digit must decrease by $1$ and the tens digit must increase by $1$ if and only if the unit digit is not $0$ hence by adding $9$ $$x+1=y-1$$ $$x-y=-2.$$ By solving both the equations we have $$x=3$$ and $$y=5$$ $$**OR**$$ The numbers whose sum of digits is equal to $8$ the numbers are $$17,26,35,44,53,62,71,80$$ and in these only $35$ is the number whose digits become equal on adding $9$

Harsh Kumar
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    Nice answer. +1 – user401699 Jan 04 '17 at 03:32
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    Are you serious? This is the same as tilper's answer!!! –  Jan 04 '17 at 03:45
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    @Rohan - There are only 13 minutes between the posting of the two answers. You should be aware that formatting your post correctly often takes more than 13 minutes. Thus Harsh Kumar may not have been aware of tilper's post when adding his own. – Paul Sinclair Jan 04 '17 at 17:55
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Let the number be $10x + y$. Adding 9 to the number will make both the digits equal. Let that digit be $m$. Then,

$$10x+y+9=11m$$ Since $x+y=8$, $$9x=11m-17$$

Since x is an integer, $$(11m-17)\%9=0$$ $$\Rightarrow 2m\%9=8$$ $$\Rightarrow m=\{4,8.5,13 \ldots \}$$

But since $m$ is a non negative integer less than 9, $$m=4$$

And therefore, original number $=44-9=35$.

nidhin
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Solving by observation and little mathematics;

let the number be $$\overline{xy}:x,y\in [1,9] \text{ and } x,y \in \text{Integers}$$

Given, $x+y=8$ and $\overline{xy} +9=\overline{zz}$

We know that, $\overline{zz} \in \{11,22,33,44,55,66,77,88,99 \} $

So, $(\overline{zz}-9=\overline{xy})\in \{11-9,22-9,33-9,44-9,55-9,66-9,77-9,88-9,99-9 \} $

or, $(\overline{zz}-9=\overline{xy})\in \{13,24,35,55-9,66-9,77-9,88-9,99-9 \} $

We can see our answer.

Sahil
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Let the $2$-digit number be $\overline{ab} = 10a + b$.

If $9$ is added to the number then both the digits become equal.

This means that $a = b$, so the number plus $9$ is $10a + a = 11a$, which is always $0 \text{ mod } {11}$ (divisible by $11$).

Therefore $\overline{ab} + 9 \equiv 0 \text{ mod } {11} \Rightarrow \overline{ab} \equiv 2 \text{ mod } {11}$.

But $a + b \equiv 8 \text{ mod } {11}$. Thus $(10a+b)-(a+b) \equiv 9a \equiv -2a \equiv 2-8 \text{ mod } {11}$, so $a \equiv 3 \text{ mod } {11}$.

As $a+b = 8$, hence $\overline{ab} = \boxed{35}$.

Toby Mak
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