Step-by-Step Solution for Number Digit Product Problem

The problem asks us to modify a given number, arrange the digits, and then find the product of specific digits in the new sequence. Let's break down the process:

1. Modifying the Even Digits

We start with the number 85624179.

First, identify the even digits in the number. The even digits are those divisible by 2.

• The even digits in 85624179 are: 8, 6, 2, 4.

Next, we add 1 to each of these even digits:

• For 8: $8 + 1 = 9$
• For 6: $6 + 1 = 7$
• For 2: $2 + 1 = 3$
• For 4: $4 + 1 = 5$

Now, replace the original even digits in the number with these new values. The odd digits remain unchanged.

• Original number: 8 5 6 2 4 1 7 9
• Modified digits: 9 5 7 3 5 1 7 9

The new sequence of digits is 9, 5, 7, 3, 5, 1, 7, 9.

2. Arranging Digits in Ascending Order

The problem states that the digits are arranged in ascending order (from smallest to largest) from left to right after the modification.

Let's sort the modified digits: 9, 5, 7, 3, 5, 1, 7, 9

Sorted sequence: 1, 3, 5, 5, 7, 7, 9, 9

3. Finding the Required Digits

We need to find two specific digits from this sorted sequence:

• Second digit from the left end: In the sequence 1, 3, 5, 5, 7, 7, 9, 9, the second digit from the left is 3.
• Third digit from the right end: Counting from the right end (9, 9, 7, 7, 5, 5, 3, 1), the third digit is 7.

Alternatively, the sorted sequence has 8 digits. The third digit from the right is the $(8 - 3 + 1) = 6$th digit from the left, which is also 7.

4. Calculating the Product

Finally, we need to find the product of these two identified digits.

• Product = (Second digit from left) $\times$ (Third digit from right)
• Product = $3 \times 7$
• Product = $21$

Therefore, the product of the digits which are second from the left end and third from the right end is 21.