Four Digit Number Properties

To find the specific four-digit number, we represent it as $d_1 d_2 d_3 d_4$, where $d_1$ is the thousands digit, $d_2$ the hundreds, $d_3$ the tens, and $d_4$ the units digit.

Digit Product Conditions

• First & Fourth Digits Product: The product of the first ($d_1$) and fourth ($d_4$) digits is 40. So, $d_1 \times d_4 = 40$.
• Middle Digits Product: The product of the middle two digits ($d_2$ and $d_3$) is 28. So, $d_2 \times d_3 = 28$.
• Digit Value Relationship: The thousands digit ($d_1$) is less than the unit digit ($d_4$), and the hundreds digit ($d_2$) is less than the tens digit ($d_3$). The amount by which $d_1$ is less than $d_4$ is equal to the amount by which $d_2$ is less than $d_3$. Mathematically: $d_4 - d_1 = d_3 - d_2$. This implies $d_1 < d_4$ and $d_2 < d_3$.

Digit Difference Calculation

1. Determine $d_1$ and $d_4$:

   From $d_1 \times d_4 = 40$, possible pairs of single digits are (5, 8) and (8, 5).

   The condition $d_1 < d_4$ means the only valid pair is $(d_1, d_4) = (5, 8)$.

2. Determine $d_2$ and $d_3$:

   From $d_2 \times d_3 = 28$, possible pairs of single digits are (4, 7) and (7, 4).

   The condition $d_2 < d_3$ means the only valid pair is $(d_2, d_3) = (4, 7)$.

3. Form the Number:

   Using the determined digits: $d_1=5, d_2=4, d_3=7, d_4=8$. The four-digit number is 5478.

4. Verify the Difference Condition:

   Check if $d_4 - d_1 = d_3 - d_2$.

   • $8 - 5 = 3$
   • $7 - 4 = 3$

   Since $3 = 3$, the difference condition is satisfied.

5. Final Answer Confirmation:

   The number 5478 meets all the specified criteria.