526 Repunits
Appendix: The multiplicative digital root
The multiplicative digital root of a number is the single digit number
obtained by iterating the digital product operation. Thus,
6244 → 6 · 2 · 4 ·4 = 192→ 1 · 9 ·2=18→ 1 cdot8=8.Theorem 18.3(Kuczma).The multiplicative digital root ofnis1ifand
only ifnis a repunit.
Proof.The digital product of a number is 1 if and only if it is a repunit.
We claim that no number can have a repunitRn,n ≥ 2 , for digital
product. If so, this would mean thatRnfactors into one digit numbers,
and its prime divisors can only be 3 and 7. In other words,Rn=3h· 7 k
for some integersh,k≥ 0.
The powers of 7 modulo 100 are 1, 7, 49, 43. Those of 3 modulo 100
are given in the first row of the table below. Their product cannot be 11
modulo 100.
1 3 9 278143298761834947412369 7 216389 677 21 63 89 67 1 3 9 27 81 43 29 87 61 83 49 47 41 23 69
49 47 41 23 69 7 21 63 89 67 1 3 9 27 81 43 29 87 61 8343 29 87 61 83 49 47 41 23 69 7 21 63 89 67 1 3 9 27 813