Difference between revisions of "2000 AIME II Problems/Problem 4"

Latest revision as of 14:55, 11 May 2022

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 $n = p_1^{e_1}p_2^{e_2} \cdots p_k^{e_k}$ is $(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)$ . If a number has $18 = 2 \cdot 3 \cdot 3$ factors, then it can have at most $3$ distinct primes in its factorization.

Dividing the greatest power of $2$ from $n$ , we have an odd integer with six positive divisors, which indicates that it either is ( $6 = 2 \cdot 3$ ) a prime raised to the $5$ th power, or two primes, one of which is squared. The smallest example of the former is $3^5 = 243$ , while the smallest example of the latter is $3^2 \cdot 5 = 45$ .

Suppose we now divide all of the odd factors from $n$ ; then we require a power of $2$ with $\frac{18}{6} = 3$ factors, namely $2^{3-1} = 4$ . Thus, our answer is $2^2 \cdot 3^2 \cdot 5 = \boxed{180}$ .

Solution 2

Somewhat similar to the first solution, we see that the number $n$ has two even factors for every odd factor. Thus, if $x$ is an odd factor of $n$ , then $2x$ and $4x$ must be the two corresponding even factors. So, the prime factorization of $n$ is $2^2 3^a 5^b 7^c...$ for some set of integers $a, b, c, ...$

Since there are $18$ factors of $n$ , we can write:

$(2+1)(a+1)(b+1)(c+1)... = 18$

$(a+1)(b+1)(c+1)... = 6$

Since $6$ only has factors from the set $1, 2, 3, 6$ , either $a=5$ and all other variables are $0$ , or $a=3$ and $b=2$ , with again all other variables equal to $0$ . This gives the two numbers $2^2 \cdot 3^5$ and $2^2 \cdot 3^2 \cdot 5$ . The latter number is smaller, and is equal to $\boxed {180}$ .

Solution 3

We see that the least number with 6 odd factors is $3^2*5$ . Multiplied by $2^2$ (as each factor of 2 doubles the odd factors, as it can be 2n or $2^2n$ . Finally, you get $180$

-dragoon

@@ Line 2: / Line 2: @@
 What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
-== Solution ==
+== Solution 1==
-{{solution}}
+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>.
+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>.
+== 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>
+-dragoon
 == See also ==
 {{AIME box|year=2000|n=II|num-b=3|num-a=5}}
+[[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

2000 AIME II (Problems • Answer Key • Resources)
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