Experiment #31

Cyclic Digits

The digits in the number 2731 have been written four times in cyclic order. That is, in each number the digits are in the same order if you move from left to right and then continue with the leftmost digit. The sum of the four digits in 2731 is 13, and 13 divides 14,443, the sum of the four numbers. Furthermore, the quotient of 14,443 divided by 13 is 1111.

Starting Points for Investigations

1. If any four-digit number is written four times in cyclic order, will the sum of its digits divide the sum of four four-digit numbers, and if so, will the quotient be 1111?

2. Investigate this situation for three-digit and five-digit numbers.

3. On the basis of your observations, form a conjecture about this property for n-digit numbers. Check your conjecture for six-digit, seven-digit, or eight-digit numbers. Do the results support your conjecture?

4. If abcd represent any four-digit number, writing each of the four numbers abcd, bcda, cdab, and dabc in expanded form and computing the sum will help you to investigate the situation. What does this show?