22. C Let $n$ be the smallest of the integers. The four consecutive integers have sum $n+n+1+n+2+n+3=4 n+6$. Then the four possible sums formed by taking three of these at a time are
$4 n+6-n=3 n+6$ (divisible by 3 so not prime)
$4 n+6-(n+1)=3 n+5$ (if $n$ is odd then this is even, so is not prime)
$4 n+6-(n+2)=3 n+4$
$4 n+6-(n+3)=3 n+3$ (divisible by 3 so not prime)
Hence we are looking for the smallest $n$ for which neither $3 n+4$ nor $3 n+5$ is prime.
\begin{tabular}{lll}
$n$ & $3 n+4$ & $3 n+5$ \\
1 & 7 prime & 8 not prime \\
2 & 10 not prime & 11 prime \\
3 & 13 prime & 14 not prime \\
4 & 16 not prime & 17 prime \\
5 & 19 prime & 20 not prime \\
6 & 22 not prime & 23 prime \\
7 & 25 not prime & 26 not prime
\end{tabular}

When $n=7$, none of the four sums is prime.
22. Jakob wrote down four consecutive positive integers. He then calculated the four possible totals made by taking three of the integers at a time. None of these totals was a prime. What is the smallest integer Jakob could have written?
A 12
B 10
C 7
D 6
E 3