Difference between revisions of "2018 AMC 10B Problems/Problem 19"
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Revision as of 02:03, 17 January 2021
- The following problem is from both the 2018 AMC 12B #14 and 2018 AMC 10B #19, so both problems redirect to this page.
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is year older than Chloe, and Zoe is exactly year old today. Today is the first of the birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
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 algebra-y)
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 if 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 .
Denote Zoe's age with , then Chloe's age is where represents Chloe's age when Zoe is one. We must have . Obviously , therefore, for 9 values of , and therefore, has factors. either takes the form of (which is too large) or . must be less than and and must be prime, therefore the only answer is . Joey's age is , which is divisible by when , therefore the answer occurs when and Joey is .
|2018 AMC 10B (Problems • Answer Key • Resources)|
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|All AMC 10 Problems and Solutions|
|2018 AMC 12B (Problems • Answer Key • Resources)|
|All AMC 12 Problems and Solutions|