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Source. The remark on p. 78 (physical p. 2 of the scan) that any homogeneous cover of $\mathbf Z\oplus\mathbf Z$ extends trivially to $\mathbf Z^n$ for every $n>2$, by reading each congruence as one in $n$ variables with all but two coefficients $0$. The paper defines a cover of $\mathbf Z^n$ only loosely there ("distinct moduli and appropriate GCD conditions on the coefficients"); the statement below makes the condition primitivity of each coefficient-and-modulus tuple. The matrix version is not stated in the paper; it and its block-matrix proof are the corpus's own addition.

T. Cochrane and G. Myerson, Covering congruences in higher dimensions, Rocky Mountain J. Math. 26 (1996), no. 1, 77–81, doi:10.1216/rmjm/1181072104; the edition read and its page mapping are named on the source card.

Statement

For every $n\ge2$, there is a finite family of primitive homogeneous linear congruences in $n$ variables, with distinct moduli greater than one, that covers $\mathbb Z^n$.

There is also a finite $n$-cover by $n\times n$ integer matrices whose determinants have pairwise distinct absolute values, all greater than one.

Proof

For each congruence

$$ax-by\equiv0\pmod m$$

in the two-dimensional homogeneous cover, use in $n$ variables the coefficient vector

$$(a,-b,0,\ldots,0).$$

Every $(x_1,\ldots,x_n)$ satisfies one of these congruences because its first two coordinates satisfy one from the original cover. Adding zero coefficients does not change the greatest common divisor with $m$, and the moduli remain distinct.

For the matrix version, take the matrices $A_j$ constructed in the two-dimensional matrix corollary and form

$$B_j=\operatorname{diag}(A_j,I_{n-2}). \tag{1}$$

Given an integer row vector $(h_1,\ldots,h_n)$, choose $j$ and an integer row vector $(k_1,k_2)$ with $(h_1,h_2)=(k_1,k_2)A_j$. Then

$$(h_1,\ldots,h_n) =(k_1,k_2,h_3,\ldots,h_n)B_j.$$

Thus the $B_j$ form an $n$-cover. Finally, $|\det B_j|=|\det A_j|$, so their determinant magnitudes remain distinct and greater than one.

Bears on. Higher-dimensional variants of covering congruences; no new one-dimensional covering-system conclusion is asserted.