Arithmetic Operation

Algebra Level 4

Consider the arithmetic operations \(\blacktriangle\) and \(\blacktriangledown\) defined by \[a\blacktriangle b= \begin{cases} a & \text{ if } \lvert a \rvert \geq \lvert b \rvert \\ b & \text{ if } \lvert a \rvert < \lvert b \rvert, \end{cases} \] \[a\blacktriangledown b= \begin{cases} a & \text{ if } \lvert a \rvert \leq \lvert b \rvert \\ b & \text{ if } \lvert a \rvert > \lvert b \rvert. \end{cases} \] How many integers \(k\) are there such that \[(-20 \blacktriangle 9) \blacktriangledown (k \blacktriangle 5)=5?\]

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