Difference between revisions of "2019 AMC 10A Problems/Problem 25"

Latest revision as of 11:32, 3 November 2024

The following problem is from both the 2019 AMC 10A #25 and 2019 AMC 12A #24, so both problems redirect to this page.

Problem

For how many integers $n$ between $1$ and $50$ , inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? (Recall that $0! = 1$ .)

$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

Solution 1

The main insight is that

$\frac{(n^2)!}{(n!)^{n+1}}$

is always an integer. This is true because it is precisely the number of ways to split up $n^2$ objects into $n$ unordered groups of size $n$ . Thus,

$\frac{(n^2-1)!}{(n!)^n}=\frac{(n^2)!}{(n!)^{n+1}}\cdot\frac{n!}{n^2}$

is an integer if $n^2 \mid n!$ , or in other words, if $\frac{(n-1)!}{n}$ , is an integer. This condition is false precisely when $n=4$ or $n$ is prime, by Wilson's Theorem. There are $15$ primes between $1$ and $50$ , inclusive, so there are $15 + 1 = 16$ terms for which

$\frac{(n^2-1)!}{(n!)^{n}}$

is potentially not an integer. It can be easily verified that the above expression is not an integer for $n=4$ as there are more factors of $2$ in the denominator than the numerator. Similarly, it can be verified that the above expression is not an integer for any prime $n=p$ , as there are more factors of p in the denominator than the numerator. Thus all $16$ values of n make the expression not an integer and the answer is $50-16=\boxed{\textbf{(D)}\ 34}$ .

SideNote: Another method to prove that $\frac{(n^2)!}{(n!)^{n+1}}$ is always an integer is instead as follows using Number Theory. Notice that $n$ will divide the numerator $n+1$ times, since $n^2 = n \times n$ contains not one but two factors of $n.$ Also, for $a < n,$ notice that $a$ divides $(n^2)!$ at least $\lfloor \frac{n^2}{a} \rfloor \ge \lfloor \frac{n^2}{n-1} \rfloor \ge \lfloor \frac{n^2 - 1}{n-1} \rfloor \ge n+1$ times. Thus, each integer from $1$ to $n$ will divide $(n^2)!$ at least $n+1$ times, which proves such a lemma. $\square{}$

Solution 2

We can use the P-Adic Valuation (more info could be found here: Mathematicial notation) of n to solve this problem (recall the P-Adic Valuation of 'n' is denoted by $v_p (n)$ and is defined as the greatest power of some prime 'p' that divides n. For example, $v_2 (6)=1$ or $v_7 (245)=2$ .) Using Legendre's formula, we know that :

$v_p (n!)= \sum_{i=1}^\infty \lfloor \frac {n}{p^i} \rfloor$

Seeing factorials involved in the problem, this prompts us to use Legendre's Formula where n is a power of a prime.

We also know that , $v_p (m^n) = n \cdot v_p (m)$ . Knowing that $a\mid b$ if $v_p (a) \le v_p (b)$ , we have that :

$n \cdot v_p (n!) \le v_p ((n^2 -1 )!)$ and we must find all n for which this is true.

If we plug in $n=p$ , by Legendre's we get two equations:

$v_p ((n^2 -1)!) = \sum_{i=1}^\infty \lfloor \frac {n^2 -1}{p^i} \rfloor = (p-1)+0+...+0 = p-1$

And we also get :

$v_p ((n!)^n) = n \cdot v_p (n!)= n \cdot \sum_{i=1}^\infty \lfloor \frac {n}{p^i} \rfloor = p \cdot ( 1+0+...0) = p$

But we are asked to prove that $n \cdot v_p (n!) \le v_p ((n^2 -1 )!) \Longrightarrow p \le p-1$ which is false for all 'n' where n is prime.

Now we try the same for $n=p^2$ , where p is a prime. By Legendre we arrive at:

$v_p ((p^4 -1)!) = p^3 + p^2 + p -3$

(as $v_p(p^4!) = p^3 + p^2 + p + 1$ and $p^4$ contains 4 factors of $p$ ) and

$p^2 \cdot v_p (p^2 !) = p^3 + p^2$

Then we get:

$p^2 \cdot v_p (p!) \le v_p ((n^4 -1)!) \Longrightarrow p^3 + p^2 \le p^3 + p^2 + p -3$ Which is true for all primes except for 2, so $2^2 = 4$ doesn't work. It can easily be verified that for all $n=p^i$ where $i$ is an integer greater than 2, satisfies the inequality : $n \cdot v_p (n!) \le v_p ((n^2 -1 )!).$

Therefore, there are 16 values that don't work and $50-16 = \boxed{\mathbf{(D)}\ 34}$ values that work.

~qwertysri987

Solution 3 (Non-Rigorous)

Notice all $15$ primes don't work as there are $n$ factors of $n$ in the denominator and $n-1$ factors of $n$ in the numerator, as easily seen through the denominator $n^n$ and the numerator $(n^2-1)!,$ which doesn't get quite to $n^2$ hence it is one short of a factor of $n$ , through $1n, 2n, \cdots, (n-1)n$ . Further experimentation finds that $n=4$ does not work as there are 11 factors of 2 in the numerator and 12 in the denominator. We also find that it seems that all other values of $n$ where $n$ is a square of a prime work. So we get $50-15-1=\boxed{34}$ and that happens to be right.

~BS2012 ~mathboy282

@@ Line 7: / Line 7: @@
 <math>\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35</math>
-==Solution==
+==Solution 1==
 The main insight is that
 <cmath>\frac{(n^2)!}{(n!)^{n+1}}</cmath>
-is always an integer. This is true because it is precisely the number of ways to split up <math>n^2</math> objects into <math>n</math> unordered groups of size <math>n</math>, which has to be an integer. Thus,
+is always an integer. This is true because it is precisely the number of ways to split up <math>n^2</math> objects into <math>n</math> unordered groups of size <math>n</math>. Thus,
-<cmath>\frac{(n^2)!}{(n!)^{n+1}}=\frac{n^2}{n!}\cdot\frac{(n^2-1)!}{(n!)^n}</cmath>
+<cmath>\frac{(n^2-1)!}{(n!)^n}=\frac{(n^2)!}{(n!)^{n+1}}\cdot\frac{n!}{n^2}</cmath>
-is not an integer if <math>n^2 \nmid n!</math>. This happens only for <math>n=4</math> or <math>n</math> prime. Thus, there are <math>16</math> integers that do not work, which means the answer is <math>\boxed{\mathbf{(D)}\ 34}</math>.
+is an integer if <math>n^2 \mid n!</math>, or in other words, if <math>\frac{(n-1)!}{n}</math>, is an integer. This condition is false precisely when <math>n=4</math> or <math>n</math> is prime, by [[Wilson's Theorem]]. There are <math>15</math> primes between <math>1</math> and <math>50</math>, inclusive, so there are <math>15 + 1 = 16</math> terms for which
+<cmath>\frac{(n^2-1)!}{(n!)^{n}}</cmath>
+is potentially not an integer. It can be easily verified that the above expression is not an integer for <math>n=4</math> as there are more factors of <math>2</math> in the denominator than the numerator. Similarly, it can be verified that the above expression is not an integer for any prime <math>n=p</math>, as there are more factors of p in the denominator than the numerator. Thus all <math>16</math> values of n make the expression not an integer and the answer is <math>50-16=\boxed{\textbf{(D)}\ 34}</math>.
+SideNote: Another method to prove that <cmath>\frac{(n^2)!}{(n!)^{n+1}}</cmath> is always an integer is instead as follows using Number Theory. Notice that <math>n</math> will divide the numerator <math>n+1</math> times, since <math>n^2 = n \times n</math> contains not one but two factors of <math>n.</math> Also, for <math>a < n,</math> notice that <math>a</math> divides <math>(n^2)!</math> at least <cmath>\lfloor \frac{n^2}{a} \rfloor \ge \lfloor \frac{n^2}{n-1} \rfloor \ge \lfloor \frac{n^2 - 1}{n-1} \rfloor \ge n+1</cmath> times. Thus, each integer from <math>1</math> to <math>n</math> will divide <math>(n^2)!</math> at least <math>n+1</math> times, which proves such a lemma. <math>\square{}</math>
+==Solution 2==
+We can use the P-Adic Valuation (more info could be found here: [[Mathematicial notation]]) of n to solve this problem (recall the P-Adic Valuation of 'n' is denoted by <math>v_p (n)</math> and is defined as the greatest power of some prime 'p' that divides n. For example, <math>v_2 (6)=1</math> or <math>v_7 (245)=2</math> .) Using Legendre's formula, we know that :
+<cmath> v_p (n!)= \sum_{i=1}^\infty   \lfloor \frac {n}{p^i} \rfloor </cmath>
+Seeing factorials involved in the problem, this prompts us to use [[Legendre's Formula]] where n is a power of a prime.
+We also know that , <math>v_p (m^n) = n \cdot v_p (m)</math> .
+Knowing that <math>a\mid b</math> if <math>v_p (a) \le v_p (b)</math> , we have that :
+<cmath> n \cdot v_p (n!) \le v_p ((n^2 -1 )!) </cmath> and we must find all n for which this is true.
+If we plug in <math>n=p</math>, by Legendre's we get two equations:
+<cmath> v_p ((n^2 -1)!) = \sum_{i=1}^\infty  \lfloor \frac {n^2 -1}{p^i} \rfloor = (p-1)+0+...+0 = p-1 </cmath>
+And we also get :
+<cmath> v_p ((n!)^n) = n \cdot v_p (n!)= n \cdot \sum_{i=1}^\infty   \lfloor \frac {n}{p^i} \rfloor = p \cdot ( 1+0+...0) = p </cmath>
+But we are asked to prove that <math> n \cdot v_p (n!) \le v_p ((n^2 -1 )!) \Longrightarrow p \le p-1 </math> which is false for all 'n' where n is prime.
+Now we try the same for <math>n=p^2</math> , where p is a prime. By Legendre we arrive at:
+<cmath>v_p ((p^4 -1)!) = p^3 + p^2 + p -3</cmath>
+(as <math>v_p(p^4!) = p^3 + p^2 + p + 1</math> and <math>p^4</math> contains 4 factors of <math>p</math>) and
+<cmath>p^2 \cdot v_p (p^2 !) = p^3 + p^2 </cmath>
+Then we get:
+<cmath> p^2 \cdot v_p (p!) \le v_p ((n^4 -1)!) \Longrightarrow p^3 + p^2 \le p^3 + p^2 + p -3 </cmath> Which is true for all primes except for 2, so <math>2^2 = 4</math> doesn't work. It can easily be verified that for all <math>n=p^i</math> where <math>i</math> is an integer greater than 2, satisfies the inequality :<cmath> n \cdot v_p (n!) \le v_p ((n^2 -1 )!).</cmath>
+Therefore, there are 16 values that don't work and <math> 50-16 = \boxed{\mathbf{(D)}\ 34}</math> values that work.
+~qwertysri987
+==Solution 3 (Non-Rigorous)==
+Notice all <math>15</math> primes don't work as there are <math>n</math> factors of <math>n</math> in the denominator and <math>n-1</math> factors of <math>n</math> in the numerator, as easily seen through the denominator <math>n^n</math> and the numerator <math>(n^2-1)!,</math> which doesn't get quite to <math>n^2</math> hence it is one short of a factor of <math>n</math>, through <math>1n, 2n, \cdots, (n-1)n</math>. Further experimentation finds that <math>n=4</math> does not work as there are 11 factors of 2 in the numerator and 12 in the denominator. We also find that it seems that all other values of <math>n</math> where <math>n</math> is a square of a prime work. So we get <math>50-15-1=\boxed{34}</math> and that happens to be right.
+~[[BS2012]]
+~mathboy282
 ==See Also==
+Video Solution by Richard Rusczyk:
+https://www.youtube.com/watch?v=9klaWnZojq0
 {{AMC10 box|year=2019|ab=A|num-b=24|after=Last Problem}}
 {{AMC12 box|year=2019|ab=A|num-b=23|num-a=25}}
 {{MAA Notice}}

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