VTMO 2026 - Grade 5 - Semifinal - SET 10 - Self-study Question 25

Quân has a special calculator with number-entry buttons and just two calculation buttons: $\times4$ and $+4$. How many natural numbers can he enter initially so that, after one or more calculation-button presses, the result is 2024? Natural numbers here include 0.

For example, both 2020 and 122 can lead to 2024:
$2020\xrightarrow{+4}2024$;
$122\xrightarrow{+4}126\xrightarrow{\times4}504\xrightarrow{\times4}2016\xrightarrow{+4}2020\xrightarrow{+4}2024$.