Find the smallest positive integers $a, b$ such that...

Question

Find the smallest positive integers $a, b$ such that:

i) $|a - b| = 3$
ii) the sum of digits of each of $a, b$ is divisible by $11$

Answer 1

Notice that it does not matter whether $a$ or $b$ is larger. For the sake of finding a solution, simply say $a + 3 = b$. Now the goal is to find:

$a$ such that $a$ and $a+3$ both have a digit sum that is divisible by $11$.

Say $a$ has decimal representation $a_ka_{k-1} \cdots a_1a_0$.

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Unless $a_0 >= 7$, $a+3$ will have a digit sum that is simply $3$ greater than the digit sum of $a$.

Therefore, $a_0$ must be $7$, $8$, or $9$.

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Now, notice that with any of $7$, $8$, or $9$ as $a_0$, the difference between the digit sums of $a$ and $a+3$ will be equal. Therefore, $a_0 = 7$ since we are seeking the smallest number possible satisfying these constraints.

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Now, we want to find the smallest number $n$ such that the difference between the digit sum of $\overbrace{9 \cdots 9}^n7$ and $1$ is divisible by $11$.

Simple trial and error shows that $n = 3$. $9997$ has a digit sum of 34, and $34-1 = 33$, which is a multiple of $11$. Therefore, $a_3 = a_2 = a_1 = 9$ and $a_0 = 7$.

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Now, notice that $a_4$ cannot be $9$ as this will increase the value of $n$.

Regardless the value of $a_4$, the digit sum of $a_4a_3a_2a_1a_0$ will not be divisible by 11. Therefore, $a_5$ must be nonzero. To ensure that $a_5$ is as small as possible, we set $a_4$ as large as possible yet less than $9$. Thus, we set $a_4 = 8$.

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Since the digit sum of $a_4a_3a_2a_1a_0$ is now equal to $42$, we know that $a_5 = 2$.

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Therefore, $a = 289997$ and $b = 290000$. This satisfies both conditions.

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Notice: While I am confident in the problem solving strategies used here, I am a high school student with no formal training in elementary number theory. This may not be the most elegant solution. I also would not discount the chance there could be a smaller solution satisfying the conditions.

Comments are definitely welcome! Let me know if I have done this correctly :)

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EDIT: Thanks to the comments, I did in fact realize that there was an issue with my assumption that $a_0 = 7$. Using the problem solving strategy exactly as before and solving for all of $a_0 = 7$, $a_0 = 8$, and $a_0 = 9$, we find that the smallest possible answer occurs when $a_0 = 9$.

Using $a_0 = 9$, we see that the smallest possible answer is, as the comments suggested, 5 digits in length.

The smallest solution therefore is $a = 89999$ and $b = 90002$.

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Thanks for the comments!