VTMO 2026 - Grade 8 - Semifinal - SET 05 - Self-study Question 11

Let $A=1^3+2^3+\cdots+12^3$. How many of these statements are true: (1) $A$ is divisible by $12$ but not $13$; (2) divisible by $13$ but not $12$; (3) a perfect square; (4) divisible by $156$; (5) divisible by $169$?