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Difference between revisions of "2019 AMC 10A Problems/Problem 11"

m (Fixed formatting)
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Solution by Aadileo
 
Solution by Aadileo
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==Solution 2==
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201 = 67 * 3. Split the answers into cases:
 +
 +
Case 1: The factor is 3^n
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Then, we would have n = 2, 3, 4, 6, 8, and 9.
 +
 +
Case 2: The factor is 67^n
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Same as case 1
 +
 +
Case 3: The factor is some combination
 +
This would be easy if we could just have any combination, as that would be simply 6*6. However, we must pair the numbers that generate squares with the numbers that generate squares and the same for cubes. In simpler terms, let's organize our n values.
 +
n = 2 is a "square" because it would be a factor of this number that is a perfect square. More generally, it is even.
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n = 3 is a "cube" because it would be a factor of this number that is a perfect cube. More generally, it is a multiple of 3.
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n = 4 is a "square"
 +
n = 6 is interesting, since its both a "square" and a "cube". Don't count this as either because it's double function would double count the numbers if we counted them simply as a "square" or as a "cube", so we will count them in another case.
 +
n = 8 is a "square"
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n = 9 is a "cube".
 +
 +
Now let's do subcases:
 +
Subcase 1: The squares are with each other.
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Since we have 3 square terms, and they would pair with 3 other square terms, we get 3*3 = 9 cases
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 +
Subcase 2: The cubes are with each other.
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Since we have 2 cube terms, and they would pair with 2 other cube terms, we get 2*2 = 4 cases
 +
 +
Subcase 3: A number pairs with n=6.
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Since any number can pair with n=6 since it's a square AND cube, there are 6 cases. Remember however that there can be two different bases (3 and 67), and they would produce different results. Thus, there are 6*2 = 12 cases
 +
 +
Thus, we add up the cases, to get 6+6+9+4+12 = 37.
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 +
iron
  
 
==See Also==
 
==See Also==

Revision as of 20:13, 11 February 2019

Problem

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?

${\textbf{(A) }32} \qquad {\textbf{(B) }36} \qquad {\textbf{(C) }37} \qquad {\textbf{(D) }39} \qquad {\textbf{(E) }41}$

Solution 1

Prime factorizing $201^9$, we get $3^9\cdot67^9$. A perfect square must have even powers of its prime factors, so our possible choices for our exponents of a perfect square are $0, 2, 4, 6, 8$ for both $3$ and $67$. This yields $5\cdot5 = 25$ perfect squares.

Perfect cubes must have multiples of $3$ for each of their prime factors' exponents, so we have either $0, 3, 6$, or $9$ for both $3$ and $67$, which yields $4\cdot4 = 16$ perfect cubes, for a total of $25+16 = 41$.

Subtracting the overcounted powers of six ($3^0\cdot67^0$ , $3^0\cdot67^6$ , $3^6\cdot67^0$, and $3^6\cdot67^6$), we get $41-4 = \boxed{\textbf{(C) }37}$.

Solution by Aadileo

Solution 2

201 = 67 * 3. Split the answers into cases:

Case 1: The factor is 3^n Then, we would have n = 2, 3, 4, 6, 8, and 9.

Case 2: The factor is 67^n Same as case 1

Case 3: The factor is some combination This would be easy if we could just have any combination, as that would be simply 6*6. However, we must pair the numbers that generate squares with the numbers that generate squares and the same for cubes. In simpler terms, let's organize our n values. n = 2 is a "square" because it would be a factor of this number that is a perfect square. More generally, it is even. n = 3 is a "cube" because it would be a factor of this number that is a perfect cube. More generally, it is a multiple of 3. n = 4 is a "square" n = 6 is interesting, since its both a "square" and a "cube". Don't count this as either because it's double function would double count the numbers if we counted them simply as a "square" or as a "cube", so we will count them in another case. n = 8 is a "square" n = 9 is a "cube".

Now let's do subcases: Subcase 1: The squares are with each other. Since we have 3 square terms, and they would pair with 3 other square terms, we get 3*3 = 9 cases

Subcase 2: The cubes are with each other. Since we have 2 cube terms, and they would pair with 2 other cube terms, we get 2*2 = 4 cases

Subcase 3: A number pairs with n=6. Since any number can pair with n=6 since it's a square AND cube, there are 6 cases. Remember however that there can be two different bases (3 and 67), and they would produce different results. Thus, there are 6*2 = 12 cases

Thus, we add up the cases, to get 6+6+9+4+12 = 37.

iron

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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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