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I apologize if this was posted already. But here goes:

On one of the

OG questions, I got the equation 29x+15y=440 for one of the statements. The coefficients are ticket prices in cents and the variables are amount bought per ticket price. Anyway, this type of equation without the whole number constraint of tickets bought would be insufficient. But since it's a whole number constraint it turns out that it can only be valid when x=10 and y=10. I actually figured it out after minutes upon minutes of racking my brain. The way I figured it out is I realized that 440-15y would be a multiple of 5, so I substituted all numbers of x that would give me a multiple of 5 that would be less than or equal to 440 and would be valid if y is an integer. Unfortunately, with a whole number constraint there isn't always just one valid combo for the 2 variables. For example, if the equation were 50x+100y=300. Then there can be either 2 tickets at 50 cents a piece and 2 tickets at $1 OR 4 tickets at 50 cents a piece and 1 ticket at $1 (not to mention if it's possible for no tickets bought at one of the 2 amounts). Is there somehow a faster way of figuring out whether only one set of x and y satisfy the equation? Would there only be 1 answer for x and y whenever the coefficients have no prime factors in common (such as 15 and 29)?

Thanks for your help.