  Let $k\ge 2$ be a positive integer and $P^+(n)$ the greatest prime factor of
a positive integer $n$ with convention $P^+(1)=1$. For any $\theta\in
\left[\frac 1{2k},\frac{17}{32k}\right)$, set
$$T_{k,\theta}(x)=\sum_{\substack{p_1\cdot\cdot\cdot p_k\le x\\
P^+(\gcd(p_1-1,...,p_k-1))\ge (p_1\cdot\cdot\cdot p_k)^\theta}}1,$$ where the
$p'$s are primes. It is proved that
$$T_{k,\theta}(x)\ll_{k}\frac{x^{1-\theta(k-1)}}{(\log x)^2},$$ which, together
with the lower bound $$T_{k,\theta}(x)\gg_{k}\frac{x^{1-\theta(k-1)}}{(\log
x)^2}$$ obtained by Wu in 2019, answer a 2015 problem of Luca, Menares and
Pizarro-Madariaga on the exact order of magnitude of $T_{k,\theta}(x)$.
  A main novelty in the proof is that, instead of using the Brun--Titchmarsh
theorem to estimate the $k^{th}$ movement of primes in arithmetic progressions,
we transform the movement to an estimation involving taking primes
simultaneously by linear shifts of primes.
