<s>[INST] Is the digital root of twin primes product larger than (3,5) always 8 
Is the digital root of twin primes product larger than $(3,5)$ always $8$?

E.g. I just checked the following and it is true was wondering if it is more widely true
$5\times 7=35$, digital root $=3+5=8$
$11\times 13=143$, digital root $=1+4+3=8$
$17\times 19=323$, digital root $=3+2+3=8$
$101\times 103=10403$, digital root $=1+0+4+0+3=8$
 [/INST] One of the basic properties of twin primes $k-1$, $k+1$ is that $k$ must be a multiple of $6$ (except $k=4$).  Set $k=6t$, and we have $$(k-1)(k+1)=(6t+1)(6t-1)=36t-1$$
Hence the product of any pair of twin primes (except $3,5$) will be congruent to $-1$ modulo $9$.  Hence the digital root of same will be $8\equiv -1\pmod{9}$.
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