Problem 324

Begin with any two-digit positive integer and multiply the two digits together. If the resulting product is a two-digit number, then repeat the process. When this process is repeated, all two-digit numbers will eventually become a single digit number. Once a product results in a single digit, the process stops.

For example,

Two-digit number 97: Step 1 $9 \times 7 = 63$, Step 2 $6 \times 3 = 18$, Step 3 $1 \times 8 = 8$. The process stops at 8 after 3 steps.

Two-digit number 48: Step 1 $4 \times 8 = 32$, Step 2 $3 \times 2 = 6$. The process stops at 6 after 2 steps.

Two-digit number 50: Step 1 $5 \times 0 = 0$. The process stops at 0 after 1 step.

(a) Beginning with the number 68, determine the number of steps required for the process to stop.

(b) Determine all two-digit numbers for which the process stops at 8 after 2 steps.

(c) Determine all two-digit numbers for which the process stops at 4.

(d) Determine a two-digit number for which the process stops after 4 steps.