Question 69
For a positive integer $n$, denote by $s(n)$ the sum of the digits of $n$ (in its usual decimal representation). For example, $s(8)=8, s(2023)=7, s(573)=15$.
a) Determine whether there exist distinct positive integers $a$ and $b$ such that
$$
2023 \cdot a + s(a) = 2023 \cdot b + s(b).
$$
b) Determine whether there exist distinct positive integers $a$ and $b$ such that
$$
a + 2023 \cdot s(a) = b + 2023 \cdot s(b).
$$
a) Determine whether there exist distinct positive integers $a$ and $b$ such that
$$
2023 \cdot a + s(a) = 2023 \cdot b + s(b).
$$
b) Determine whether there exist distinct positive integers $a$ and $b$ such that
$$
a + 2023 \cdot s(a) = b + 2023 \cdot s(b).
$$