Parasitic Number - Derivation

Derivation

An n-parasitic number can be derived by starting with a digit k (which should be equal to n or greater) in the rightmost (units) place, and working up one digit at a time. For example, for n = 4 and k = 7:

4•7=28

4•87=348

4•487=1948

4•9487=37948

4•79487=317948

4•179487=717948.

So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.

Notice that the repeating decimal

Thus

In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that k ≥ n, and take the period of the repeating decimal k/(10n−1). This will be where m is the length of the period; i.e. the multiplicative order of 10 modulo (10n − 1).

For another example, if n = 2, then 10n − 1 = 19 and the repeating decimal for 1/19 is

So that for 2/19 is double that:

The length m of this period is 18, the same as the order of 10 modulo 19, so 2 × (1018 − 1)/19 = 105263157894736842.

105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.