Question 36
Let $r$, $t$ and $a$ be three positive integers such that $r \le t$ and let
$$
A = \{1, 2, 3, \dots, r\}, \quad B = \{a+r+1, a+r+2, a+r+3, \dots, a+r+t\}.
$$
For any set $P$ of integers and any integer $x$, we define $P + x = \{p + x : p \in P\}$, called the translate of $P$ by $x$.
a. If $r = t$, prove that $Z$ (the set of all integers) is a union of a certain class of mutually disjoint translates of $A \cup B$ if and only if $a = kr$ for some positive integer $k$.
b. If $r \lt t$, prove that $Z$ is a union of a certain class of mutually disjoint translates of $A \cup B$ if and only if $a = k(r + t)$ for some positive integer $k$.
$$
A = \{1, 2, 3, \dots, r\}, \quad B = \{a+r+1, a+r+2, a+r+3, \dots, a+r+t\}.
$$
For any set $P$ of integers and any integer $x$, we define $P + x = \{p + x : p \in P\}$, called the translate of $P$ by $x$.
a. If $r = t$, prove that $Z$ (the set of all integers) is a union of a certain class of mutually disjoint translates of $A \cup B$ if and only if $a = kr$ for some positive integer $k$.
b. If $r \lt t$, prove that $Z$ is a union of a certain class of mutually disjoint translates of $A \cup B$ if and only if $a = k(r + t)$ for some positive integer $k$.