When a certain number is multiplied by 7, the product entirely comprises ones only (1111...). What is the smallest such number ?

(a) 15713
(b) 15723
(c) 15783
(d) 15873 ✓ UPSC's answer

Why the answer is (d)

• The problem requires finding the smallest integer $N$ such that $7 \times N$ results in a number consisting entirely of 1s (a repunit), i.e., $7N = \underbrace{11\dots1}_{k}$.

• This implies $N = \frac{11\dots1}{7}$. We test repunits of increasing length to find the first one divisible by 7.

• For $k=1$, $1/7$ is not an integer. For $k=2$, $11/7$ is not an integer. For $k=3$, $111/7$ is not an integer. For $k=4$, $1111/7$ is not an integer. For $k=5$, $11111/7$ is not an integer. For $k=6$, $111111/7$ is not an integer.

• For $k=7$, $1111111 \div 7 = 15873$. Since 15873 is an integer, this is the smallest such number.

• Therefore, the smallest number is 15873, which corresponds to option (d).

Why the other options are wrong

(a) 15713: 15713 multiplied by 7 equals 109991, which contains zeros and nines, not entirely ones.
(b) 15723: 15723 multiplied by 7 equals 110061, which contains zeros and sixes, not entirely ones.
(c) 15783: 15783 multiplied by 7 equals 110481, which contains zeros, fours, and eights, not entirely ones.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2021, held on 10 October 2021. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.