Question 15
Let $N$ be the number of ordered triples $(A, B, C)$ of integers satisfying the conditions
• $0 \le A \lt B \lt C \le 99,$
• there exist integers $a,$ $b,$ and $c,$ and prime $p$ where $0 \le b \lt a \lt c \lt p,$
• $p$ divides $A - a,$ $B - b,$ and $C - c,$ and
• each ordered triple $(A, B, C)$ and each ordered triple $(b, a, c)$ form arithmetic sequences.
Find $N.$
• $0 \le A \lt B \lt C \le 99,$
• there exist integers $a,$ $b,$ and $c,$ and prime $p$ where $0 \le b \lt a \lt c \lt p,$
• $p$ divides $A - a,$ $B - b,$ and $C - c,$ and
• each ordered triple $(A, B, C)$ and each ordered triple $(b, a, c)$ form arithmetic sequences.
Find $N.$