#1132 · Number Theory · Divisibility

Repeating‑digit divisor

Let $n$ be an odd positive integer. Form a decimal number by writing $n$ digits $d_{1} d_{2} \dots d_{n}$ and then repeating this block, obtaining $d_{1} d_{2} \dots d_{n} d_{1} d_{2} \dots d_{n}$ . Prove that this $2 n$ ‑digit number is divisible both by the $n$ ‑digit number $d_{1} d_{2} \dots d_{n}$ and by the number $9090 \dots 9091$ which has $n - 1$ digits (the pattern $90$ repeated $\frac{n - 1}{2}$ times followed by a final $1$ ).

~10 minProof · checked by AIrepeating blocksmodular arithmetic

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VII. Scales of Notation, Examples VII.b, No. 40

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