#Problem 16
Prove any perfect number ,except 6, has digital root 1.
Prove any perfect number ,except 6, has digital root 1.
Solution scheme and approach:
We already know that what is digital root(look under concept of number theory) and we also learned that dividing any number by 9 we get digital root.
Let, the number is
dkdk-1dk-2…….d2d1Now,
dkdk-1dk-2…….d2As 26k=1(mod 9) and 26k+1-1=1(mod 9) we can write,
26k(26k+1-1) =1(mod 9)….(ii)
Case 2: p=6k+5
As 26k+4= 24(mod 9)=7(mod 9) and 26k+5-1=25-1(mod 9)= 4 we can write,
26k+4(26k+5-1) =7*4(mod 9)=1(mod 9)……….(iii)
From (i),(ii) and (iii) we
can conclude any perfect number ,without 6, will be having digital root
1.(Q.E.D.)
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