At most $\left[\frac{17\times 22}{3\times 5}\right]=24$ pieces of cardboard can be cut
The problem is whether the maximum can be achieved
And here it is
[asy][asy]
unitsize(5mm);
int i, j;
for(i=0; i<23; i=i+1){
for(j=0; j<18; j=j+1){
draw((i,j));}}
draw((0,0)--(0,17));
draw((3,0)--(3,5));
draw((5,5)--(5,17));
draw((6,0)--(6,5));
draw((9,0)--(9,5));
draw((10,5)--(10,17));
draw((12,0)--(12,12));
draw((13,12)--(13,17));
draw((16,12)--(16,17));
draw((17,0)--(17,12));
draw((19,12)--(19,17));
draw((22,0)--(22,17));
draw((0,0)--(22,0));
draw((12,3)--(22,3));
draw((12,6)--(22,6));
draw((12,9)--(22,9));
draw((10,12)--(22,12));
draw((0,5)--(12,5));
draw((0,8)--(10,8));
draw((0,11)--(10,11));
draw((0,14)--(10,14));
draw((0,17)--(22,17));
[/asy][/asy]