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Consider repdigit numbers made entirely of the digit 8, i.e. 8, 88,…
Question
Consider repdigit numbers made entirely of the digit 8, i.e. 8, 88, 888, 8888, and so on. Find the number of digits in the smallest such number that is divisible by 99.
✓ Verified answer: 18checked by our engine — not a guess
Step-by-step solution
The k-digit repdigit of 8's equals 8·R_k where R_k = (10^k − 1)/9 is the repunit of length k.
Since gcd(8,99)=1, we need 99 | R_k.
Now 99 = 9·11.
Divisibility of R_k by 9: the digit sum of R_k is k, so 9 | R_k iff 9 | k.
Divisibility by 11: alternating digit sum of R_k is 0 if k even and 1 if k odd, so 11 | R_k iff k is even.
The smallest k that is a multiple of 9 and even is lcm(9,2) = 18.
Final answer18
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