# Sunday Times Teaser 2677 – One For Each Day

### by Danny Roth

George created a thosand digit number by repeating the same non-zero digit one thousand times. He then replaced its first and last digits with another non-zero digit to create a thousand digit number with only two different digits.

Martha commented that the resulting number was divisible by seven. George added that it was actually divisible by exactly seven of 2, 3, 4, 5, 6, 7, 8, 9 and 11.

What were the first two digits of this number?

As far as I can see, there is something wrong with this teaser. Either the number has seven of the given divisors, NOT including 7, or it has just six, in which case there are two solutions.

It would seem that George’s last comment should have contradicted Martha’s rather than adding to it. But the relevant part of the Sunday Times description of the teaser (which I use above) implies that the number is divisible by 7 and by exactly seven of (2, 3, 4, 5, 6, 7, 8, 9, 11).

This gives:

277..772 == 0 mod 7 and has 6 of the divisors (2, 3, 4, 6, 7, 11).
633..336 == 1 mod 7 and has 7 of the divisors (2, 3, 4, 6, 8, 9, 11).
877..778 == 0 mod 7 and has 6 of the divisors (2, 3, 6, 7, 9, 11).

I have discussed this puzzle with Victor Bryant (who oversees these teasers for the Sunday Times) and he confirms that there is an error in this teaser.

The intended answer is 277..772 but the author mistakenly thought that this number is divisible by 8, giiving seven divisors when there are only six.

• I didn’t really have a problem solving this puzzle (as you see from my solution below). I interpreted the statement “George added that it was actually …” as being George correcting Martha’s statement, so it didn’t really matter if Martha’s statement was true or false, all we needed to do was find a number divisible by exactly seven of the given divisors. Which is what my code does, and it finds a unique solution of 6333…3336.

Of course if the setter of the puzzle really did intend the answer to be 2777…7772 then it is indeed a flawed puzzle.

2. Python’s support for large integers means this one can be solved in a straight-forward way.

3. In a similar fashion to Jim’s solution, I let Python manage the large numbers, but I still don’t understand the strange wording of the Teaser.

Transcript of messages to/from the ST…

Renny Barrett 13 Jan
to puzzle.feedback
Hi there,

I just tried to solve this teaser and found it impossible – on searching the web, I came across this page which came to the same conclusion that I did: https://cgi.gladman.plus.com/wp/?page_id=1797

Is there something amiss in the wording of the Teaser, please?

Thanks,

Renny

Victor Bryant
16:30 (1 hour ago)
to me

Dear Renny,

Thanks for the e-mail (forwarded to me) concerning Teaser 2677. You are, of course, quite right that (due to a slip) there is no answer to the Teaser as stated. I am very sorry for the inconvenience.

Many thanks for the interest and for writing to us.

Best wishes,
Victor Bryant
(Teaser editor)