# Need Help USACO Silver Multiples of 2019 external sol

can someone explain matthew chen solution
https://usaco.guide/problems/ac-MultipleOf2019/user-solutions

1. i don’t understand what this number means
d_k + 10d_{k+1} + 100d_{k+2} + … + 10^(n-k)d_n
2. why do we multiple by 10^(k-1)?
3. and how does multiplying number above by 10^(k-1) turns into this?
S_n - S_{k-1} = 0 (mod 2019) <=> S_n = S_{k-1} (mod 2019)
4. also, what does S_{k-1} mean?
5. how is this true? 10^(-1) = 202 (mod 2019)
isn’t this 0.1 = 202 (mod 2019)? not sure how they got 202
6. why do we do this " multiply subsequent terms by a factor of 202 to form an equivalent sequence for S_n in the field F_{2019}."
7. can someone give a better summary than mines?
1. take mod of substring
1. multiply remainder from 1. by factor than take mod again
• Q: why do multiply by factor, isn’t that redundant?
• why not just use the remainder directly as a hash key?
• 3.use remainder from 2. as a key for hash residues
1. calculate answer from hash residues