Difference between revisions of "2018 AMC 10B Problems/Problem 19"
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<cmath>\frac{c+n}{n+1} = c - \frac{(c-1)(n)}{n+1} = c - (c-1)\frac{n}{n+1}</cmath> | <cmath>\frac{c+n}{n+1} = c - \frac{(c-1)(n)}{n+1} = c - (c-1)\frac{n}{n+1}</cmath> | ||
We want <math>c-1</math> to be divisible by nine values of <math>n-1,</math> and the least possible number with nine factors is <math>2^2 \cdot 3^2 = 36.</math> So <math>c-1 = 36, c = 37,</math> and Joey's age is <math>37+1 = 38.</math> We repeat the process, and we seek the second smallest <math>n</math> for which | We want <math>c-1</math> to be divisible by nine values of <math>n-1,</math> and the least possible number with nine factors is <math>2^2 \cdot 3^2 = 36.</math> So <math>c-1 = 36, c = 37,</math> and Joey's age is <math>37+1 = 38.</math> We repeat the process, and we seek the second smallest <math>n</math> for which | ||
− | <cmath>\frac{ | + | <cmath>\frac{38+n}{n+1} = 38 - \frac{(37)(n)}{n+1} = 38 - (37)\frac{n}{n+1}</cmath> |
is an integer. This occurs for <math>n=36,</math> and our answer is thus <math>38+36 = 74,</math> and <math>7+4=\boxed{(\textbf{E}) 11}</math> | is an integer. This occurs for <math>n=36,</math> and our answer is thus <math>38+36 = 74,</math> and <math>7+4=\boxed{(\textbf{E}) 11}</math> | ||
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==Video Solution== | ==Video Solution== |
Revision as of 13:08, 10 August 2021
- The following problem is from both the 2018 AMC 12B #14 and 2018 AMC 10B #19, so both problems redirect to this page.
Contents
Problem
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?
Solution 1
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
Solution 2
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 .
Solution 3
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 .
Solution 4
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 .
Solution 5
Note that Zoe's age years from now is and Chloe's age is for some positive integer. We want there to be nonnegative integers such that divides In other words, is an integer for nine values of We can rewrite this as We want to be divisible by nine values of and the least possible number with nine factors is So and Joey's age is We repeat the process, and we seek the second smallest for which is an integer. This occurs for and our answer is thus and
Video Solution
~savannahsolver
Video Solution
https://youtu.be/zfChnbMGLVQ?t=111
~ pi_is_3.14
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.