Lots of people found the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones. The answer is 15873 since 7 x 15873 = 111111.

The people who explained how they got this answer by working systematically using division were Martin, Richard, David and James of Hethersett High School, Norwich; Chris and Adrian of Poltair Community School, St Austell, Cornwall; Prav and Sheli from the North London Collegiate School Maths Puzzle Club; Maren and Corinna of Camden School for Girls and Soh from Raffles Institution, Singapore..

The best account was from Amy of Madras College, St Andrews.
Well done Amy! She first defined a Repunit as a number consisting
entirely of 1's so that R _{1} = 1; R _{2} = 11; R
_{3} = 111; R _{4} = 1111 etc.

Investigating by division:

11 / 7 = 1.571...

111 / 7 = 15.857...

1111 / 7 = 158.714...

11111 / 7 = 11587.28...

111111 / 7 = 15873.

So 15873 is the smallest whole number which, when multiplied by 7, gives a repunit.

Extending the question to find what other numbers will also work, we know 3 must be the smallest number to work because a number ending in a 1 will be odd so it will not divide by 2.

11 / 3 = 3.666...

111 / 3 = 37 .

So 37 is the smallest whole number which, when multiplied by 3, gives a repunit.

The number 4 will not work for the same reason that 2 does not work, and 6 won't work, neither will 8.. Any number multiplied by 5 will end in a 0 or 5, so 5 won't work, so 9 will be the next number to try.

11 / 9 = 1.222...

R_{3}/ 9 = 12. 33...

R_{4}/ 9 = 123.44...

R_{5}/9 = 1234.55...

R_{6}/ 9 = 12345.66...

R_{7}/ 9 = 123456.77...

R_{8}/ 9 = 1234567.88...

R_{9}/9 = 123456789

So 123456789 is the smallest whole number which, when multiplies by 9, gives a repunit.