## 30. New Mathematics

My son has learned to count in a numeral system different from the decimal system, and instead of 136, he writes 253 in this system. In what numeral system is my son counting?
30. Let $a$ be the base of an unknown numeral system. When my son writes "253" in this system, it represents $2 a^{2}+5 a+3$; according to the problem, this corresponds to the number 136 in the decimal system. Therefore, we can write

$$
2 a^{2}+5 a+3=136
$$

i.e.

$$
2 a^{2}+5 a-133=0
$$

or

$$
(a-7)(2 a+19)=0
$$

Since $a$ is a positive integer, the only solution to this equation is 7. My son was using the septenary (base-7) system. 31.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-191.jpg?height=288&width=397&top_left_y=467&top_left_x=835)