### The ‘$4$’ button on my calculator is spoilt, so I cannot enter numbers which contain the digit $4$. Moreover, my calculator does not display the digit $4$ if $4$ is part of an answer either. Thus I cannot enter the calculation $3 \times 24$ and do not attempt to do so. Also, the result of multiplying $3$ by $18$ is displayed as $5$ instead of $54$ and the result of multiplying $6$ by $8$ is displayed as $8$ instead of $48$. If I multiply a positive one-digit number by a positive two-digit number on my calculator and it displays $26$, how many possibilities could I have multiplied?

Answer:

$6$

Step by Step Explanation:
1. The possible results when a positive $1-$digit number is multiplied by a positive $2-$digit number that will display $26$ is $26, 426, 246, \text{ and } 264$.
2. \begin{align} & 26 = 2 \times 13, 1 \times 26 \\ & 426 = 6 \times 71\\ & 246 = 3 \times 82\\ & 264 = 3 \times 88, 8 \times 33 \end{align}
3. Hence, there are $6$ such possibilities.

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