Problem
NT-B1-M10-P021 A Multiple Made of Ones
#21
★★★★☆ Level 4 of 5
Let \(\gcd(m,10)=1\). Prove that there exists a number consisting only of digit \(1\) that is divisible by \(m\).
Consider \(R_1,\ldots,R_m\) and their residues modulo \(m\).
If one of \(R_1,\ldots,R_m\) is divisible by \(m\), we are done. Otherwise two residues coincide: \(R_i\equiv R_j\pmod m\), \(i
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