Difference between revisions of "2018 AMC 10A Problems/Problem 19"
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We see that the unit digit of <math>3^x</math>, for some integer <math>x</math>, will only be <math>1</math> when <math>x</math> is a multiple of <math>4</math>. Now, let's count how many numbers in <math>B</math> are divisible by <math>4</math>. This can be done by simply listing: | We see that the unit digit of <math>3^x</math>, for some integer <math>x</math>, will only be <math>1</math> when <math>x</math> is a multiple of <math>4</math>. Now, let's count how many numbers in <math>B</math> are divisible by <math>4</math>. This can be done by simply listing: | ||
<cmath>2000,2004,2008,2012,2016.</cmath> | <cmath>2000,2004,2008,2012,2016.</cmath> | ||
− | There are <math>5</math> numbers in <math>B</math> divisible by <math>4</math> out of the <math>2018-1999+1=20</math> total numbers. Therefore, the probability that <math>3</math> is picked from <math>A</math> and a number divisible by <math>4</math> is picked from <math>B</math> is <math>\frac{1}{5}\cdot \frac{5}{20}=\frac{1}{20}</math> | + | There are <math>5</math> numbers in <math>B</math> divisible by <math>4</math> out of the <math>2018-1999+1=20</math> total numbers. Therefore, the probability that <math>3</math> is picked from <math>A</math> and a number divisible by <math>4</math> is picked from <math>B</math> is <math>\frac{1}{5}\cdot \frac{5}{20}=\frac{1}{20}.</math> |
Similarly, we can look at the repeating units digit for <math>7</math>: | Similarly, we can look at the repeating units digit for <math>7</math>: | ||
<cmath>7\cdot 7=9</cmath> | <cmath>7\cdot 7=9</cmath> | ||
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We see that the unit digit of <math>7^y</math>, for some integer <math>y</math>, will only be <math>1</math> when <math>y</math> is a multiple of <math>4</math>. This is exactly the same conditions as our last case with <math>3</math> so the probability of this case is also <math>\frac{1}{20}</math>. | We see that the unit digit of <math>7^y</math>, for some integer <math>y</math>, will only be <math>1</math> when <math>y</math> is a multiple of <math>4</math>. This is exactly the same conditions as our last case with <math>3</math> so the probability of this case is also <math>\frac{1}{20}</math>. | ||
Since <math>5\cdot 5=25</math> and <math>25</math> ends in <math>5</math>, the units digit of <math>5^w</math>, for some integer, <math>w</math> will always be <math>5</math>. Thus, the probability in this case is <math>0</math>. | Since <math>5\cdot 5=25</math> and <math>25</math> ends in <math>5</math>, the units digit of <math>5^w</math>, for some integer, <math>w</math> will always be <math>5</math>. Thus, the probability in this case is <math>0</math>. | ||
− | The last case we need to consider is when the number <math>9</math> is chosen from <math>A</math>. This happens with probability <math>\frac{1}{5}</math> | + | The last case we need to consider is when the number <math>9</math> is chosen from <math>A</math>. This happens with probability <math>\frac{1}{5}.</math> We list out the repeating units digit for <math>9</math> as we have done for <math>3</math> and <math>7</math>: |
<cmath>9\cdot 9=1</cmath> | <cmath>9\cdot 9=1</cmath> | ||
<cmath>1\cdot 9=9</cmath> | <cmath>1\cdot 9=9</cmath> | ||
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~Nivek | ~Nivek | ||
+ | ~very minor edits by virjoy2001 | ||
== Solution 2 == | == Solution 2 == |
Revision as of 22:39, 28 November 2020
Problem
A number is randomly selected from the set , and a number is randomly selected from . What is the probability that has a units digit of ?
Solution 1
Since we only care about the units digit, our set can be turned into . Call this set and call set . Let's do casework on the element of that we choose. Since , any number from can be paired with to make have a units digit of . Therefore, the probability of this case happening is since there is a chance that the number is selected from . Let us consider the case where the number is selected from . Let's look at the unit digit when we repeatedly multiply the number by itself: We see that the unit digit of , for some integer , will only be when is a multiple of . Now, let's count how many numbers in are divisible by . This can be done by simply listing: There are numbers in divisible by out of the total numbers. Therefore, the probability that is picked from and a number divisible by is picked from is Similarly, we can look at the repeating units digit for : We see that the unit digit of , for some integer , will only be when is a multiple of . This is exactly the same conditions as our last case with so the probability of this case is also . Since and ends in , the units digit of , for some integer, will always be . Thus, the probability in this case is . The last case we need to consider is when the number is chosen from . This happens with probability We list out the repeating units digit for as we have done for and : We see that the units digit of , for some integer , is only when is an even number. From the numbers in , we see that exactly half of them are even. The probability in this case is Finally, we can add all of our probabilities together to get
~Nivek ~very minor edits by virjoy2001
Solution 2
Since only the units digit is relevant, we can turn the first set into . Note that for all odd digits , except for 5. Looking at the second set, we see that it is a set of all integers between 1999 and 2018. There are 20 members of this set, which means that, , this set has 5 values which correspond to , making the probability equal for all of them. Next, check the values for which it is equal to . There are values for which it is equal to 1, remembering that only if , which it is not. There are 20 values in total, and simplifying gives us or .
Solution 3
By Euler's Theorem, we have that , iff . Hence , work. Also note that because , and the latter mod 10 is clearly 1. So , work(not counting multiples of 4 as we would be double counting if we did). We can also note that because , and by the same logic as why , we are done. Hence , and work(not counting any of the aforementioned cases as that would be double counting). We cannot make anymore observations that add more with units digit , hence the number of that have units digit one is . And the total number of combinations of an element of the set of all and an element of the set of all is . Hence the desired probability is , which is answer choice . ~vsamc
Video Solution
~IceMatrix
See Also
2018 AMC 10A (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 |
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