Discussion
When a certain number is multiplied by 56, the product entirely comprises of digit 8's only (888...). Which of the following is the smallest such number?
Answer: C
The smallest such number is 15873.
The product must be a number made only of 8s, so it is 8 times a repunit (111...1). Since \(56 = 8 \times 7\), we need the smallest repunit divisible by 7, and the number asked for is that repunit divided by 7.
\[ \begin{aligned} 111111 &= 7 \times 15873 \\ 15873 \times 56 &= 888888 \end{aligned} \]
No shorter repunit (1, 11, 111, 1111, 11111) is divisible by 7, so 888888 is the first all-8 product and 15873 the smallest multiplier. A quick check of the others: 15093, 15738 and 16283 times 56 end in 8 but contain other digits, for example \(15738 \times 56 = 881328\).
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