Problem
NT-B2-M11-P013 A Repeated Three-Digit Block
#13
★★★★☆ Level 4 of 5
Let \(a\ne0\), and let \(N=\overline{abcabc}\). Prove that \(N\) is divisible by \(7\), \(11\), and \(13\).
1001 Problems in Classical Number Theory (method inspiration) · Problem 302
Factor out the common three-digit block \(\overline{abc}\).
We have \(N=1000\overline{abc}+\overline{abc}=1001\overline{abc}\). But \(1001=7\cdot11\cdot13\). Therefore \(N\) is divisible by \(7\), \(11\), and \(13\).
It looks elementary, but the same idea later works for blocks and moduli \(10^k\pm1\).