The following puzzle first appeared in February 1999.
Choose any three decimal digits a, b, c, not necessarily distinct, and form the nine-digit number abc,abc,abc. Divide this number by 333,667. Was the quotient a whole number? Can you explain why?
Choose any four decimal digits a, b, c, d, not necessarily distinct, and form the eight-digit number abcdabcd. Divide this number by 137. Was the quotient a whole number? Can you explain why?
Choose any five decimal digits a, b, c, d, e, not necessarily distinct, and form the 10-digit number abcdeabcde. Divide this number by 9091. Was the quotient a whole number? Can you explain why?
Golomb's Answers:
The nine-digit number abc,abc,abc can be written as abc x 1,001,001 where 1,001,001 = 3 x 333,667. Hence, abc,abc,abc divided by 333,667 is the whole number 3 x abc.
We have abcdabcd = abcd x 10001 = 137 x 73 x abcd. Hence abcdabcd divided by 137 is 73 x abcd, a whole number.
Similarly, abcdeabcde = abcd x 100,001 where 100,001 = 11 x 9091. Hence abcdeabcde divided by 9091 is the whole number 11 x abcde.
