r+z+o= 36
o:z= 2:5=> no of obsidian gem=o= 2k
No. of zirconium gems=z= 5k
total no of o & z= 7k
r+7k=36
k=\(\frac{(36-r)}{7}\)
r=8 => k=\(\frac{28}{7}\) =4
r= 15=>k=\(\frac{24}{7}\) =3
r=22=> k=\(\frac{14}{7}\) =2
r=29=> k=\(\frac{7}{7}\) 1
r=12=> k= not an integer.
Hence\(r \neq{12}\)
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o:z= 2:5=> no of obsidian gem=o= 2k
No. of zirconium gems=z= 5k
total no of o & z= 7k
r+7k=36
k=\(\frac{(36-r)}{7}\)
r=8 => k=\(\frac{28}{7}\) =4
r= 15=>k=\(\frac{24}{7}\) =3
r=22=> k=\(\frac{14}{7}\) =2
r=29=> k=\(\frac{7}{7}\) 1
r=12=> k= not an integer.
Hence\(r \neq{12}\)
...