Problem
NT-B1-M12-P023 Set 5. Infinitely Many Multiples
#23
★★★★☆ Level 4 of 5
Prove that there are infinitely many numbers consisting only of digits \(0\) and \(1\) that are divisible by \(2027\).
First build one multiple of ones, then append zeros on the right.
Since \(\gcd(2027,10)=1\), there exists a repunit \(R_s\) divisible by \(2027\). Then for every \(t\ge0\), the number \(R_s\cdot10^t\) is also divisible by \(2027\) and consists of ones followed by zeros. Different \(t\) give infinitely many different numbers.
Strengthens the construction: not one number, but an infinite family.