Parasitic number

An n parasitic number (in base 10) is a positive natural number which, when multiplied by n, results in movement of the last digit of its decimal representation to its front. Here n is itself a single digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one place. For example:
4 × 128205 = 512820, so 128205 is 4 parasitic. Most mathematicians do not allow leading zeros to be used, and that is a commonly followed convention. So even though 4 × 25641 = 102564, the number 25641 is not 4 parasitic.

The step by step derivation algorithm depicted above is a great core technique but will not find all n parasitic numbers. It will get stuck in an infinite loop when the derived number equals the derivation source. An example of this occurs when n = 5 and k = 5. The 42 digit n parasitic number to be derived is 102040816326530612244897959183673469387755. Check the steps in Table One below. The algorithm begins building from right to left until it reaches step 15—then the infinite loop occurs. Lines 16 and 17 are pictured to show that nothing changes. There is a fix for this problem, and when applied, the algorithm will not only find all n parasitic numbers in base ten, it will find them in base 8 and base 16 as well. Look at line 15 in Table Two. The fix, when this condition is identified and the n parasitic number has not been found, is simply to not shift the product from the multiplication, but use it as is, and append n (in this case 5) to the end. After 42 steps, the proper parasitic number will be found.

1. 5 × 5 = 25 − Shift = 55 2. 5 × 55 = 275 − Shift = 755 3. 5 × 755 = 3775 − Shift = 7755 4. 5 × 7755 = 38775 − Shift = 87755 5. 5 × 87755 = 438775 − Shift = 387755 6. 5 × 387755 = 1938775 − Shift = 9387755 7. 5 × 9387755 = 46938775 − Shift = 69387755 8. 5 × 69387755 = 346938775 − Shift = 469387755 9. 5 × 469387755 = 2346938775 − Shift = 3469387755 10. 5 × 3469387755 = 17346938775 − Shift = 73469387755 11. 5 × 73469387755 = 367346938775 − Shift = 673469387755 12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755 13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755 14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755 15. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755 16. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755 17. 5 × 183673469387755 = 918367346938775 − Shift = 183673469387755

1. 5 × 5 = 25 − Shift = 55 2. 5 × 55 = 275 − Shift = 755 3. 5 × 755 = 3775 − Shift = 7755 4. 5 × 7755 = 38775 − Shift = 87755 5. 5 × 87755 = 438775 − Shift = 387755 6. 5 × 387755 = 1938775 − Shift = 9387755 7. 5 × 9387755 = 46938775 − Shift = 69387755 8. 5 × 69387755 = 346938775 − Shift = 469387755 9. 5 × 469387755 = 2346938775 − Shift = 3469387755 10. 5 × 3469387755 = 17346938775 − Shift = 73469387755 11. 5 × 73469387755 = 367346938775 − Shift = 673469387755 12. 5 × 673469387755 = 3367346938775 − Shift = 3673469387755 13. 5 × 3673469387755 = 18367346938775 − Shift = 83673469387755 14. 5 × 83673469387755 = 418367346938775 − Shift = 183673469387755 15. 5 × 183673469387755 = 918367346938775 − Shift = 9183673469387755 16. 5 × 9183673469387755 = 45918367346938775 − Shift = 59183673469387755 17. 5 × 59183673469387755 = 295918367346938775 − Shift = 959183673469387755
There is one more condition to be aware of when working with this algorithm, leading zeros must not be lost. When the shift number is created it may contain a leading zero which is positionally important and must be carried into and through the next step. Calculators and computer math methods will remove leading zeros. Look at Table Three below displaying the derivation steps for n = 4 and k = 4. The Shift number created in step 4, 02564, has a leading zero which is fed into step 5 creating a leading zero product. The resulting Shift is fed into Step 6 which displays a product proving the 4 parasitic number ending in 4 is 102564.

1. 4 × 4 = 16 − Shift = 64 2. 4 × 64 = 256 − Shift = 564 3. 4 × 564 = 2256 − Shift = 2564 4. 4 × 2564 = 10256 − Shift = 02564 5. 4 × 02564 = 010256 − Shift = 102564 6. 4 × 102564 = 410256 − Shift = 102564

In general, if we relax the rules to allow a leading zero, then there are 9 n parasitic numbers for each n. Otherwise only if k ≥ n then the numbers do not start with zero and hence fit the actual definition. Other n parasitic integers can be built by concatenation. For example, since 179487 is a 4 parasitic number, so are 179487179487, 179487179487179487 etc.