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GCD & LCMhard
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For which of the following linear Diophantine equations do integer solutions exist? (Recall: ax+by=cax + by = cax+by=c has integer solutions iff gcd⁡(a,b)∣c\gcd(a,b) \mid cgcd(a,b)∣c.)

I. 6x+10y=126x + 10y = 126x+10y=12 II. 6x+10y=156x + 10y = 156x+10y=15 III. 15x+25y=2015x + 25y = 2015x+25y=20 IV. 15x+25y=10015x + 25y = 10015x+25y=100