Problem
NT-B1-M10-P011 Divisibility of \(R_6\)
#11
★★☆☆☆ Level 2 of 5
Prove that \(R_6=111111\) is divisible by \(37\).
Use \(10^3\equiv1\pmod{37}\).
Since \(10^3=1000\equiv1\pmod{37}\), we have \(10^6\equiv1\pmod{37}\). Hence \(37\mid 10^6-1\). The number \(9\) is coprime to \(37\), so from \(R_6=\frac{10^6-1}{9}\) it follows that \(37\mid R_6\).
Shows how to derive divisibility of a repunit from a power of \(10\).