235. Another curious example of multiplication. Here is another puzzle from Professor Rackbrain.

What number, when multiplied by $18, 27, 36, 45, 54, 63, 72$, 81 or 99, gives a product whose first and last digits match the corresponding digits of the multiplier, and when multiplied by 90, gives a product whose last two digits match the digits of the multiplier?
235. The desired number is 987654321, which when multiplied by 18 gives 17777777778 with 1 and 8 at the beginning and end, respectively. The same is true for other multipliers, except for 90, when we get 88888888890 with 90 at the end.

[The author did not notice numbers like 1001, 10101, and 100101, composed of 0 and 1, with 1 at the ends and not containing two consecutive 1s, each of which is also a solution to the problem. \(-M . \Gamma\).