Question

The product $\left(8\right)\times (888...8)$ where the second factor has $k$ digits, is an integer whose digits have a sum of $1000.$ The value of $k$ is

• A. $911$
• B. $919$
• C. $991$
• D. $999$

Answer 1

The correct option is C $991$

We list down the first few products for $k=1,2,3,\cdots$

For $k=1,$ the product is $8\times 8=64$
For $k=2,$ the product is $8\times 88=704$
For $k=3,$ the product is $8\times 888=7104$
For $k=4,$ the product is $8\times 8888=71104$
For $k=5,$ the product is $8\times 88888=711104$

Observing the pattern, we can infer that the products have digits in the form $7,(k-2)1s,04$ as we go from left to right.
So, $7+(k-2)+0+4=1000$
$\Rightarrow k=991$