The opposite of a two-digit number, both different from zero, is the number obtained by swapping the order of its digits. For example, the opposite of 25 is 52 and the opposite of 79 is 97. Which of the numbers below is not the sum of a two-digit number and its opposite?
A) 44
B) 99
C) 121
D) 165
E) 181
Solution

## ALTERNATIVE E

Let $n$ be a two-digit number, with $a$ being its tens digit and $b$ its units digit; then $n=10a+b$. If $a$ and $b$ are both different from zero, the reverse of $n$ is $10b+a$. Thus, the sum of $n$ and its reverse is:

$$
(10a+b)+(10b+a)=11a+11b=11(a+b)
$$

and, therefore, the sum of a number and its reverse is always a multiple of 11. It is enough to note that all the options are multiples of 11, except for 181.

One can also verify that the other options are all sums of a number and its reverse; indeed, $44=13+31, 99=18+81, 121=29+92$, and $165=69+96$.

To explain how these expressions were found, let's take $165=11 \times 15$ as an example. The initial reasoning shows that if we choose non-zero digits $a$ and $b$ such that their sum is 15, then 165 will be the sum of the number $10a+b$ and its reverse. For example, we can take $a=6$ and $b=9$; for this choice, we get the expression $165=69+96$. Other choices are possible; for example, $a=8$ and $b=7$ leads to $165=87+78$. The same reasoning applies to the other alternatives.