AMC 10 Daily Practice - Diophantine Equations

, . . Plug in and solve for , we have .

Let the prices of chicken eggs, duck eggs, and quail eggs be , , and dollars, respectively. We have

Also,

Then, .

Let Joey's age be , Chloe's age be , and we know that Zoe's age is . We know that there must be values such that where is an integer.Therefore, and . Therefore, we know that, as there are solutions for , there must be solutions for . We know that this must be a perfect square. Testing perfect squares, we see that , so . Therefore, . Now, since , by similar logic, , so and Joey will be and the sum of the digits is .

Here's a different way of saying the above solution:

If a number is a multiple of both Chloe's age and Zoe's age, then it is a multiple of their difference. Since the difference between their ages does not change, then that means the difference between their ages has factors. Therefore, the difference between Chloe and Zoe's age is , so Chloe is , and Joey is . The common factor that will divide both of their ages is , so Joey will be . .

Similar approach to above, just explained less concisely and more in terms of the problem (less algebray),

Let denote Chloe's age, denote Joey's age, and denote Zoe's age, where is the number of years from now. We are told that is a multiple of exactly nine times. Because is at and will increase until greater than , it will hit every natural number less than , including every factor of . For to be an integral multiple of , the difference must also be a multiple of , which happens iff is a factor of . Therefore, has nine factors. The smallest number that has nine positive factors is (we want it to be small so that Joey will not have reached three digits of age before his age is a multiple of Zoe's). We also know and . Thus, By our above logic, the next time is a multiple of will occur when is a factor of . Because is prime, the next time this happens is at , when . .