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Source. The second of the two constructions described before Lemma 2, p. 79 (physical p. 3 of the scan); the five-class cover it starts from is the introductory example on p. 77. The proof below is written here and is complete relative to the external existence input stated below. It is materially different from the explicit twenty-class example in Lemma 2.

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

Assume the existence of one finite covering system

$$\{(a_i,m_i):1\le i\le r\}$$

with distinct moduli all greater than $12$. Then there is a composite covering system with distinct moduli: every modulus is composite, but the family still covers all integers.

The source cites Section F13 of Richard K. Guy's 1981 Unsolved Problems in Number Theory for the required one-dimensional cover. That cited construction is an explicit external input here; its proof and residue classes are not reconstructed from this paper.

Proof

The introductory five-class cover is

$$(0,2),(0,3),(1,4),(1,6),(11,12). \tag{1}$$

It covers every integer: even integers lie in $(0,2)$; among odd integers, those that are $1$ modulo $4$ lie in $(1,4)$, while residues $3$, $7$, and $11$ modulo $12$ lie respectively in $(0,3)$, $(1,6)$, and $(11,12)$.

Apply the affine map $y\mapsto2y+1$ to (1). It shows that

$$S=\{(1,4),(1,6),(3,8),(3,12),(23,24)\} \tag{2}$$

covers every odd integer. Indeed, $y\equiv a\pmod m$ implies $2y+1\equiv2a+1\pmod {2m}$.

Apply instead $y\mapsto2y$ to the external cover. The family

$$T=\{(2a_i,2m_i):1\le i\le r\} \tag{3}$$

covers every even integer. Its moduli are distinct composite integers and are all greater than $24$. The five moduli in (2) are the distinct composite integers $4,6,8,12,24$. Therefore no modulus in $S$ occurs in $T$, and $S\cup T$ is a composite cover with distinct moduli.

Relation to the minimum-modulus problem

The input is the existence of one cover with distinct moduli whose minimum modulus exceeds $12$, taken from Guy's book and not proved in the paper. The paper says nothing about whether such covers exist for every lower bound on the minimum modulus, the question of Problem 2.

Bears on. The supply of composite covers used by the homogeneous lifting method, with the stated external limitation.