Finding a missing digit

Extract

‘There is a simple argument which shows that, for any positive real number x, the digit 7 appears in the decimal representation of one of the numbers x, 2x, … , 79x. I spent a wet Saturday afternoon showing that 79 can be reduced to 42, which is best possible.’

Some thirty years ago this interesting problem was given to me verbally by David Masser, who was a reader in mathematics at Nottingham University. I solved the problem, over several days, and forgot about it. While tidying up papers in retirement I found the notes on the problem, and when I mentioned this to David he too had forgotten about it until he searched through his notes. He now encourages me to publicise the problem, which is presented here with a small extension.