Question Number 226117 by Spillover last updated on 20/Nov/25
Answered by Spillover last updated on 20/Nov/25
Answered by Spillover last updated on 20/Nov/25
Answered by Spillover last updated on 20/Nov/25
$$ \\ $$A = 111111…11
The base ten Rep unit is given by: An = (10ⁿ – 1)/9, where n is the number of 1's
The base ten Rep unit is given by: An = (10ⁿ – 1)/9, where n is the number of 1's
Test: 11 = (10² – 1)/9
111 = (10³ – 1)/9
Now, A = 111111…11 (2025 1's) = (10²⁰²⁵ – 1)/9
Now, 2025 × A = 2025•(10²⁰²⁵ – 1)/9 = 225(10²⁰²⁵ – 1)
= 225•10²⁰²⁵ – 225
Let get the general form for 225.10ⁿ – 225
Let's consider when n≥5,
225•10⁵ – 225 = 22499775
225•10⁶ – 225 = 224999775
225•10⁷ – 225 = 2249999775
Notice that, we have a pattern, and that the number of 9's is three less than the value of n
Now, when n = 2025, the number of 9's = 2022
So, 2025 × A = 225•10²⁰²⁵ – 225 = 224999…999775 (number of 9's = 2022)
Total sum of digits of the product, 2025 × A = (2 + 2 + 4 + 2022(9) + 7 + 7 + 5)
= 18225
*: . Total sum of digits of the product, 2025 × A = 18225