3 [ Fundamental Theorem of Arithmetic. Factorization into Prime Factors]

Authors: Pukas Yu.O., Fedumkin L.E.

Yura wrote down a four-digit number. Lenya added 1 to the first digit of this number, 2 to the second, 3 to the third, and 4 to the fourth, and then multiplied the resulting sums. Lenya got 234. What number could Yura have written down?

#
Note that $234=2 \cdot 3 \cdot 3 \cdot 13$. The maximum factor obtained by Lёnya could be $9+4=13$. This factor was certainly present in the product, because smaller numbers are not divisible by 13, but 234 is. Therefore, the last digit of Yura's number is nine. The other three factors give a product of 18. There cannot be a one among them, as numbers do not start with zero. The only possible variant is $18=2 \cdot 3 \cdot 3$. It is clear that the third digit in this case is 0, and for the first two digits, there are two possible variants.

## Answer

2009 or 1109.