Difference between revisions of "1995 AIME Problems/Problem 6"

Latest revision as of 07:14, 4 November 2022

Problem

Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n_{}$ but do not divide $n_{}$ ?

Solution 1

We know that $n^2 = 2^{62}3^{38}$ must have $(62+1)\times (38+1)$ factors by its prime factorization. If we group all of these factors (excluding $n$ ) into pairs that multiply to $n^2$ , then one factor per pair is less than $n$ , and so there are $\frac{63\times 39-1}{2} = 1228$ factors of $n^2$ that are less than $n$ . There are $32\times20-1 = 639$ factors of $n$ , which clearly are less than $n$ , but are still factors of $n$ . Therefore, using complementary counting, there are $1228-639=\boxed{589}$ factors of $n^2$ that do not divide $n$ .

Solution 2

Let $n=p_1^{k_1}p_2^{k_2}$ for some prime $p_1,p_2$ . Then $n^2$ has $\frac{(2k_1+1)(2k_2+1)-1}{2}$ factors less than $n$ .

This simplifies to $\frac{4k_1k_2+2k_1+2k_2}{2}=2k_1k_2+k_1+k_2$ .

The number of factors of $n$ less than $n$ is equal to $(k_1+1)(k_2+1)-1=k_1k_2+k_1+k_2$ .

Thus, our general formula for $n=p_1^{k_1}p_2^{k_2}$ is

Number of factors that satisfy the above $=(2k_1k_2+k_1+k_2)-(k_1k_2+k_1+k_2)=k_1k_2$

Incorporating this into our problem gives $19\times31=\boxed{589}$ .

Solution 3

Consider divisors of $n^2: a,b$ such that $ab=n^2$ . WLOG, let $b\ge{a}$ and $b=\frac{n}{a}$

Then, it is easy to see that $a$ will always be less than $b$ as we go down the divisor list of $n^2$ until we hit $n$ .

Therefore, the median divisor of $n^2$ is $n$ .

Then, there are $(63)(39)=2457$ divisors of $n^2$ . Exactly $\frac{2457-1}{2}=1228$ of these divisors are $<n$

There are $(32)(20)-1=639$ divisors of $n$ that are $<n$ .

Therefore, the answer is $1228-639=\boxed{589}$ .

Video Solution by OmegaLearn

https://youtu.be/jgyyGeEKhwk?t=259

~ pi_is_3.14

@@ Line 2: / Line 2: @@
 Let <math>n=2^{31}3^{19}.</math>  How many positive [[integer]] [[divisor]]s of <math>n^2</math> are less than <math>n_{}</math> but do not divide <math>n_{}</math>?
-== Solution ==
+== Solution 1 ==
-We know that <math>n^2</math> must have <math>63\times 39</math> [[factor]]s by its [[prime factorization]]. There are <math>\frac{63\times 39-1}{2} = 1228</math> factors of <math>n^2</math> that are less than <math>n</math>, because if they form pairs <math>a</math>, then there is one factor per pair that is less than <math>n</math>. There are <math>32\times20-1 = 639</math> factors of <math>n</math> that are less than <math>n</math> itself. These are also factors of <math>n^2</math>. Therefore, there are <math>1228-639=\boxed{589}</math> factors of <math>n</math> that do not divide <math>n</math>.
+We know that <math>n^2 = 2^{62}3^{38}</math> must have <math>(62+1)\times (38+1)</math> [[factor]]s by its [[prime factorization]]. If we group all of these factors (excluding <math>n</math>) into pairs that multiply to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} = 1228</math> factors of <math>n^2</math> that are less than <math>n</math>. There are <math>32\times20-1 = 639</math> factors of <math>n</math>, which clearly are less than <math>n</math>, but are still factors of <math>n</math>. Therefore, using complementary counting, there are <math>1228-639=\boxed{589}</math> factors of <math>n^2</math> that do not divide <math>n</math>.
+== Solution 2 ==
+Let <math>n=p_1^{k_1}p_2^{k_2}</math> for some prime <math>p_1,p_2</math>. Then <math>n^2</math> has <math>\frac{(2k_1+1)(2k_2+1)-1}{2}</math> factors less than <math>n</math>.
+This simplifies to <math>\frac{4k_1k_2+2k_1+2k_2}{2}=2k_1k_2+k_1+k_2</math>.
+The number of factors of <math>n</math> less than <math>n</math> is equal to <math>(k_1+1)(k_2+1)-1=k_1k_2+k_1+k_2</math>.
+Thus, our general formula for <math>n=p_1^{k_1}p_2^{k_2}</math> is
+Number of factors that satisfy the above <math>=(2k_1k_2+k_1+k_2)-(k_1k_2+k_1+k_2)=k_1k_2</math>
+Incorporating this into our problem gives <math>19\times31=\boxed{589}</math>.
+== Solution 3 ==
+Consider divisors of <math>n^2: a,b</math> such that
+<math>ab=n^2</math>.
+WLOG, let <math>b\ge{a}</math> and <math>b=\frac{n}{a}</math>
+Then, it is easy to see that <math>a</math> will always be less than <math>b</math> as we go down the divisor list of <math>n^2</math> until we hit <math>n</math>.
+Therefore, the median divisor of <math>n^2</math> is <math>n</math>.
+Then, there are <math>(63)(39)=2457</math> divisors of <math>n^2</math>. Exactly <math>\frac{2457-1}{2}=1228</math> of these divisors are <math><n</math>
+There are <math>(32)(20)-1=639</math> divisors of <math>n</math> that are <math><n</math>.
+Therefore, the answer is <math>1228-639=\boxed{589}</math>.
+== Video Solution by OmegaLearn ==
+https://youtu.be/jgyyGeEKhwk?t=259
+~ pi_is_3.14
 == See also ==
@@ Line 9: / Line 44: @@
 [[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

1995 AIME (Problems • Answer Key • Resources)
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