Difference between revisions of "2000 AIME II Problems/Problem 4"
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What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | ||
− | == Solution == | + | == Solution 1== |
− | We use the fact that the number of divisors of a number <math>n = p_1^{e_1}p_2^{e_2} \cdots p_k^{e_k}</math> is <math>(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)</math>. If a number has <math>18 = 2 \cdot | + | We use the fact that the number of divisors of a number <math>n = p_1^{e_1}p_2^{e_2} \cdots p_k^{e_k}</math> is <math>(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)</math>. If a number has <math>18 = 2 \cdot 3 \cdot 3</math> factors, then it can have at most <math>3</math> distinct primes in its factorization. |
Dividing the greatest power of <math>2</math> from <math>n</math>, we have an odd integer with six positive divisors, which indicates that it either is (<math>6 = 2 \cdot 3</math>) a prime raised to the <math>5</math>th power, or two primes, one of which is squared. The smallest example of the former is <math>3^5 = 243</math>, while the smallest example of the latter is <math>3^2 \cdot 5 = 45</math>. | Dividing the greatest power of <math>2</math> from <math>n</math>, we have an odd integer with six positive divisors, which indicates that it either is (<math>6 = 2 \cdot 3</math>) a prime raised to the <math>5</math>th power, or two primes, one of which is squared. The smallest example of the former is <math>3^5 = 243</math>, while the smallest example of the latter is <math>3^2 \cdot 5 = 45</math>. | ||
Suppose we now divide all of the odd factors from <math>n</math>; then we require a power of <math>2</math> with <math>\frac{18}{6} = 3</math> factors, namely <math>2^{3-1} = 4</math>. Thus, our answer is <math>2^2 \cdot 3^2 \cdot 5 = \boxed{180}</math>. | Suppose we now divide all of the odd factors from <math>n</math>; then we require a power of <math>2</math> with <math>\frac{18}{6} = 3</math> factors, namely <math>2^{3-1} = 4</math>. Thus, our answer is <math>2^2 \cdot 3^2 \cdot 5 = \boxed{180}</math>. | ||
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+ | == Solution 2 == | ||
+ | Somewhat similar to the first solution, we see that the number <math>n</math> has two even factors for every odd factor. Thus, if <math>x</math> is an odd factor of <math>n</math>, then <math>2x</math> and <math>4x</math> must be the two corresponding even factors. So, the prime factorization of <math>n</math> is <math>2^2 3^a 5^b 7^c...</math> for some set of integers <math>a, b, c, ...</math> | ||
+ | |||
+ | Since there are <math>18</math> factors of <math>n</math>, we can write: | ||
+ | |||
+ | <math>(2+1)(a+1)(b+1)(c+1)... = 18</math> | ||
+ | |||
+ | <math>(a+1)(b+1)(c+1)... = 6</math> | ||
+ | |||
+ | Since <math>6</math> only has factors from the set <math>1, 2, 3, 6</math>, either <math>a=5</math> and all other variables are <math>0</math>, or <math>a=3</math> and <math>b=2</math>, with again all other variables equal to <math>0</math>. This gives the two numbers <math>2^2 \cdot 3^5</math> and <math>2^2 \cdot 3^2 \cdot 5</math>. The latter number is smaller, and is equal to <math>\boxed {180}</math>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | |||
+ | We see that the least number with 6 odd factors is <math>3^2*5</math>. Multiplied by <math>2^2</math> (as each factor of 2 doubles the odd factors, as it can be 2n or <math>2^2n</math>. Finally, you get <math>180</math> | ||
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+ | -dragoon | ||
== See also == | == See also == | ||
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 14:55, 11 May 2022
Contents
[hide]Problem
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
Solution 1
We use the fact that the number of divisors of a number is . If a number has factors, then it can have at most distinct primes in its factorization.
Dividing the greatest power of from , we have an odd integer with six positive divisors, which indicates that it either is () a prime raised to the th power, or two primes, one of which is squared. The smallest example of the former is , while the smallest example of the latter is .
Suppose we now divide all of the odd factors from ; then we require a power of with factors, namely . Thus, our answer is .
Solution 2
Somewhat similar to the first solution, we see that the number has two even factors for every odd factor. Thus, if is an odd factor of , then and must be the two corresponding even factors. So, the prime factorization of is for some set of integers
Since there are factors of , we can write:
Since only has factors from the set , either and all other variables are , or and , with again all other variables equal to . This gives the two numbers and . The latter number is smaller, and is equal to .
Solution 3
We see that the least number with 6 odd factors is . Multiplied by (as each factor of 2 doubles the odd factors, as it can be 2n or . Finally, you get
-dragoon
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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