Question 141
Consider the set $X=\{1,2,3, \ldots, 2018\}$. How many positive integers $k$ with $2 \leq k \leq 2017$ that satisfy the following conditions:
i) There exists some partition of the set $X$ into $1009$ disjoint pairs which are $(a_{1}, b_{1}), (a_{2}, b_{2}), \ldots, (a_{1009}, b_{1009})$ with $|a_{i}-b_{i}| \in \{1, k\}$.
ii) For all partitions satisfy the condition (i), the sum $T=\sum_{i=1}^{1009}|a_{i}-b_{i}|$ has the right most digit is $9$.
i) There exists some partition of the set $X$ into $1009$ disjoint pairs which are $(a_{1}, b_{1}), (a_{2}, b_{2}), \ldots, (a_{1009}, b_{1009})$ with $|a_{i}-b_{i}| \in \{1, k\}$.
ii) For all partitions satisfy the condition (i), the sum $T=\sum_{i=1}^{1009}|a_{i}-b_{i}|$ has the right most digit is $9$.