Let $k\ge 2$ be a fixed integer. We consider sums of type $\sum_{ n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, taken over the $k$-dimensional spherical region $\{ (n_1,\ldots,n_k)\in { \Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x\}$, where $F:{ \Bbb Z}^k\to { \Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations $\sum_{ n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$ and $\sum_{ n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, involving the GCD and LCM of the integers $n_1,\ldots,n_k$, where $f:{ \Bbb N}\to { \Bbb C}$ belongs to certain classes of functions.
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