Correct Option
The problem requires identifying the smallest integer which, when multiplied by 7, results in a product composed exclusively of the digit '1'. Let the unknown number be 'x' and the product be 'P'. According to the condition, P = 7 × x, where P is a number consisting solely of ones (e.g., 1, 11, 111, etc.).
To find the smallest 'x', we must determine the smallest 'P' (composed entirely of ones) that is perfectly divisible by 7. We systematically test numbers formed by repeating the digit '1':
- 1 ÷ 7 yields a remainder.
- 11 ÷ 7 yields a remainder.
- 111 ÷ 7 yields a remainder.
- 1111 ÷ 7 yields a remainder.
- 11111 ÷ 7 yields a remainder.
- 111111 ÷ 7 = 15873 with no remainder.
Therefore, 111111 is the smallest number composed entirely of ones that is perfectly divisible by 7. The corresponding smallest number 'x' is 111111 / 7 = 15873.
Incorrect Options
Options 15713, 15723, and 15783 are incorrect because they do not satisfy the given condition. When multiplied by 7, these numbers produce results that are not composed entirely of the digit '1':
- 15713 × 7 = 109991
- 15723 × 7 = 110061
- 15783 × 7 = 110481
These products contain digits other than '1', thereby failing to meet the problem's criteria. The number 15873 is the unique smallest integer satisfying the condition.