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In mathematics, the 6 repeated digits rule is a unique arithmetic property involving 6 digit numbers with all of its digits being the same value. There are 9 numbers that share these properties, which are:
- 111,111
- 222,222
- 333,333
- 444,444
- 555,555
- 666,666
- 777,777
- 888,888
- 999,999
The 6 repeated digits rule states that each one of these 9 numbers is divisible by the factors of 7, 11, and 13. This property is especially interesting because of the rarity of the factors 7, 11, and 13, the uniqueness of the structure of the 9 numbers to which the property can be applied, and the nature of the repetition found in this rule.
Proof
editThe reason that the 6 repeated digits rule works is because of the factors of the number 111,111. The number 111,111 itself is divisible by 7, 11, and 13.
- 111,111 ÷ 7 = 15,873
- 111,111 ÷ 11 = 10,101
- 111,111 ÷ 13 = 8,547
Because this is true, each one of the 9 numbers shown above will share the same factors because of the fact that they are all multiples of 111,111. For example, the number 444,444 is 111,111 × 4, meaning that it too will share the same 3 factors as 111,111, which are 7, 11, and 13. This same idea can be applied for each of the 9 numbers above, which will all share the same 3 factors:
- 111,111 × 1 = 111,111
- 111,111 × 2 = 222,222
- 111,111 × 3 = 333,333
- 111,111 × 4 = 444,444
- 111,111 × 5 = 555,555
- 111,111 × 6 = 666,666
- 111,111 × 7 = 777,777
- 111,111 × 8 = 888,888
- 111,111 × 9 = 999,999
It is important to note that the inverse is not true. While the numbers 111,111 and 444,444 both share the factors of 7, 11, and 13, the number 444,444 shares an additional factor of 2. The same is true for numbers such as 888,888 and 999,999, which have additional factors of 4 and 3 respectively. The rule simply states that these 9 numbers share the factors of 7, 11 and 13.
Cyclic properties
editThe 6 repeated digits rule has a connection to cyclic numbers, numbers which repeat their digits when multiplied by certain values. The number 999,999, when divided by 7, will result in the number 142,857, which is the most famous natural cyclic number. The cycle is shown below:
- 142857 × 1 = 142857
- 142857 × 2 = 285714
- 142857 × 3 = 428571
- 142857 × 4 = 571428
- 142857 × 5 = 714285
- 142857 × 6 = 857142[1]