Problem
NT-B1-M10-P022 A Number Made of Nines
#22
★★★★☆ Level 4 of 5
Is it true that for every positive integer \(m\), there exists a number written only with digit \(9\) that is divisible by \(m\)? Give the exact corrected statement.
Check what happens for \(m=2\) and \(m=5\).
A number made of nines is always odd and ends in \(9\), so it is not divisible by \(2\) or \(5\). Thus the original statement is false. The corrected statement is: if \(\gcd(m,10)=1\), then such a number exists. Indeed, by the previous construction there is \(R_n\) divisible by \(m\); then \(9R_n\) consists only of nines and is also divisible by \(m\).
Intentional trap checking coprimality conditions.