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Decimal Numbers of Concatenated Blocks
Which are Fractional Numbers

Consider the decimal representations of 2/7 and 3/13

2/7 = 0.285714285714…
3/13 = 0.230769230769…

These are concatenated blocks of 285714 and 230769. Some questions immediately arise, such as

Well, for starters the digit sums of 285714 and 230769 are both 9 which indicates that they are multiples of 9. Their ratios when divided by 9 are 31746 and 25641 respectively.

As to the second question consider the following:

2/285714 = 0.000007000007…
3/2307669 = 0.000013000013…

But also

2/3 = 0.6666…
2/9 = 0.2222…

And less obviously

1/1 = 0.9999…

Note the product of the reciprocal of the fraction on the LHS with its concatenation block;

(7/2)(285714) = 999999
(13/3)(230769) = 999999
(285714/2)(7) = 999999
(230769/3)(13) = 999999
(3/2)(6) = 9
9(1) = 9
1(9) = 9

Proposition: Let (p/q) be a fraction less than or equal to 1 and K be the concatenation block. Then (q/p)K is equal to 99…99, where the number of 9's in this product is equal to m, the number digits in K.

Proof:

The equation

p/q = 0.KKK…
means
p/q = (K/10m) + (K/102m) + (K/103m) +…
or, equivalently
p/q = (K/10m)[1 + (1/10m) + (1/10m)² + (1/10m)³ + …

On the right there is the geometric series sum whose value is equal to 1/(1−1/10m). Therefore

p/q = (K/10m)[1/(1−1/10m).]
and when the denominators on
the right are multiplied together
p/q = K/(10m−1)
(q/p)K = 10m−1

The term (10m−1) is just m 9's in a row.

The concatenation block is arbitrary to a degree. For example, the concatenation block for 0.333333… could be chosen as 3, 33, 333 or any number of 3's in a row. The proposition holds for any such choice.

The Set of Integers such that their
decimal representations of their reciprocals
have the concatenated block form

The product qK is equal to p(99…99). If p and q have no factor in common, q must be a factor of 99…99. But 99…99 equal to 9(11…11) so q must be a factor 9 or 11…11. For m equal to 1, q must be a factor of 9. The factors of 9 are 1, 3 and 9. So there :not be any other fractions with a denominator other than 1, 3 and 9 whose decimal representation of the concatenated block That is to say the fractions less than one and in reduced form which have a block concatenation form are 1/3, 2/3, 1/9, 2/9, 4/9, 5/9, 7/9 and 8/9.

for m equal to 2 the only integers with the property being considered are the divisors of 99 which are 1, 3, 9, 11, 33 and 99.

For m=3 the integers 1, 3 and 9 have the property but also the divisors of 111. The factorization of 111 is 3*37 so its divisors are 1, 3, 37, 111, which means that they have the property but also 333 and 999.

The factorization of 1111 is 11*101 which means its divisors are 1, 11, 101 and 1111. So the set of numbers with the property is {1, 3, 9, 11, 33, 99, 101, 1111}.

The factorization of 111111 is 3 * 7 * 11 * 13 * 37, so its divisors include 1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 111, 143, 231, 259, 407 , 429, 481, 777, 1001, 1221, 1443, 2849, 3003, 3367, 5291, 8547, 10101, 15873, 37037 and 111111. It becomes tedious to list all the divisors of 111111 . but it may include more besides the ones shown. The divisors of 999999 include all of the divisiors of 111111 and their multiples by 9.

Conclusion

The fractions (p/q) which have a concatenated m-digit block form are the such that q is a divisor of 10m−1=99…99 and p is any integer which does not have a factor in common with q.


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