R, 50 bytes

k=scan();while(any(utf8ToInt(c(F<-F+k,""))>49))0;F 

Try it online!

Recall that the ASCII codepoints of 0 and 1 are 48 and 49.

The while statement goes through all multiples of k, converting each to character with c(…,"") and checking whether any codepoints are greater than 49, i.e. whether there any digits greater than 1.