Analyzing Digit Modifications in 21673458

The question asks us to modify the digits of the number 21673458 based on whether they are odd or even, and then find the product of the greatest and smallest digits in the resulting number.

Step-by-step Solution

1. Identify the digits in the original number: 2, 1, 6, 7, 3, 4, 5, 8.

2. Separate the digits into odd and even groups:

   • Odd digits: 1, 7, 3, 5
   • Even digits: 2, 6, 4, 8

3. Apply the specified modifications:

   • Add 1 to each odd digit:
     • $1 + 1 = 2$
     • $7 + 1 = 8$
     • $3 + 1 = 4$
     • $5 + 1 = 6$
   • Subtract 1 from each even digit:
     • $2 - 1 = 1$
     • $6 - 1 = 5$
     • $4 - 1 = 3$
     • $8 - 1 = 7$

4. Construct the new number using the modified digits in their original positions: The sequence of modified digits is 1, 2, 5, 8, 4, 3, 6, 7. The new number is 12584367.

5. Identify the greatest and smallest digits in the new number (12584367):

   • Smallest digit: 1
   • Greatest digit: 8

6. Calculate the product of the greatest and smallest digits:

   Product = Smallest digit $\times$ Greatest digit

   Product = $1 \times 8 = 8$

The product of the greatest and smallest digit in the new number is 8.