## Task 5 - 170735

Determine all two-digit numbers for which both the following statement (1) and the following statement (2) are true:

(1) If you insert the digit 5 between the unit digit and the tens digit of the two-digit number, you get a number that is exactly 230 greater than the original number.

(2) If you place the digit 5 in front of the two-digit number, you get an integer multiple of the original number.
Assuming a two-digit number satisfies the conditions of the problem, it is at least 10 and at most 99: consequently, after increasing it by 230, a number is formed that is at least 240 and at most 329. Of these numbers, only 250, 251, 252, 253, 254, 255, 256, 257, 258, and 259 have a 5 as the tens digit.

Therefore, only the numbers 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 can satisfy condition (1). They satisfy this condition because inserting the digit 5 as mentioned in (1) replaces the tens digit 2 with the sequence 25, thereby increasing each number by 230.

If you place the digit 5 in front of each of them, you get the numbers 520, 521, 522, 523, 524, 525, 526, 527, 528, and 529. These are each 500 greater than the original number. Therefore, a number formed in this way is an integer multiple of the original number if and only if 500 is also an integer multiple of it. This is true for the numbers 20 and 25 among those that satisfy (1). Therefore, these are all the two-digit numbers that satisfy both conditions.