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Difference between revisions of "2023 AMC 10A Problems/Problem 23"

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==Problem==
 
==Problem==
−
If the positive integer <math>c</math> has positive integer divisors <math>a</math> and <math>b</math> with <math>c = ab</math>, then <math>a</math> and <math>b</math> are said to be <math>\textit{complementary}</math> divisors of <math>c</math>. Suppose that <math>N</math> is a positive integer that has one complementary pair of divisors that differ by <math>20</math> and another pair of complementary divisors that differ by <math>23</math>. What is the sum of the digits of <math>N</math>?
+
If the positive integer <math>n</math> has positive integer divisors <math>a</math> and <math>b</math> with <math>n = ab</math>, then <math>a</math> and <math>b</math> are said to be <math>\textit{complementary}</math> divisors of <math>n</math>. Suppose that <math>N</math> is a positive integer that has one complementary pair of divisors that differ by <math>20</math> and another pair of complementary divisors that differ by <math>23</math>. What is the sum of the digits of <math>N</math>?
  
 
<math>\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19</math>
 
<math>\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19</math>
  
 
== Solution 1 ==
 
== Solution 1 ==
−
Consider positive <math>a, b</math> with a difference of <math>20</math>. Suppose <math>b = a-20</math>. Then, we have that <math>(a)(a-20) = c</math>. If there is another pair of two integers that multiply to 30 but have a difference of 23, one integer must be greater than <math>a</math>, and one must be smaller than <math>a-20</math>. We can create two cases and set both equal. We have <math>(a)(a-20) = (a+1)(a-22)</math>, and <math>(a)(a-20) = (a+2)(a-21)</math>. Starting with the first case, we have <math>a^2-20a = a^2-21a-22</math>,or <math>0=-a-22</math>, which gives <math>a=-22</math>, which is not possible. The other case is <math>a^2-20a = a^2-19a-42</math>, so <math>a=42</math>. Thus, our product is <math>(42)(22) = (44)(21)</math>, so <math>c = 924</math>. Adding the digits, we have <math>9+2+4 = \boxed{\textbf{(C) } 15}</math>.
+
Consider positive integers <math>a, b</math> with a difference of <math>20</math>. Suppose <math>b = a-20</math>. Then, we have <math>(a)(a-20) = n</math>. If there is another pair of two integers that multiply to <math>n</math> but have a difference of 23, one integer must be greater than <math>a</math>, and the other must be smaller than <math>a-20</math>. We can create two cases and set both equal.  
−
-Sepehr2010
+
 
 +
We have <math>(a)(a-20) = (a+1)(a-22) \text{or} (a+2)(a-21).</math> Note that if we go further to <math>(a+3)(a-20)</math> and beyond, that would violate the condition that one of the two integers must be smaller than <math>a-20.</math>
 +
 
 +
Starting with the first case, we have <math>a^2-20a = a^2-21a-22</math>,or <math>0=-a-22</math>, which gives <math>a=-22</math>, which is not possible. The other case is <math>a^2-20a = a^2-19a-42</math>, so <math>a=42</math>. Thus, our product is <math>(42)(22) = (44)(21)</math>, so <math>c = 924</math>. Adding the digits, we have <math>9+2+4 = \boxed{\textbf{(C) } 15}</math>.
 +
 
 +
-Sepehr2010, mathboy282, the_eaglercraft_grinder
  
 
==Solution 2 ==
 
==Solution 2 ==
Line 27: Line 32:
 
x^2+20x&=y^2+23y\\
 
x^2+20x&=y^2+23y\\
 
4x^2+4\cdot20x &= 4y^2+4\cdot23y\\
 
4x^2+4\cdot20x &= 4y^2+4\cdot23y\\
−
4x^2+4\cdot20x+20^2-20^2 &= 4y^2+4\cdot23y+23^3-23^2\\
+
4x^2+4\cdot20x+20^2-20^2 &= 4y^2+4\cdot23y+23^2-23^2\\
 
(2x+20)^2-20^2 &= (2y+23)^2-23^2\\
 
(2x+20)^2-20^2 &= (2y+23)^2-23^2\\
 
23^2-20^2 &= (2y+23)^2-(2x+20)^2\\
 
23^2-20^2 &= (2y+23)^2-(2x+20)^2\\
Line 37: Line 42:
 
<cmath>\begin{align}
 
<cmath>\begin{align}
 
129 or 43 &= (2y+2x+43)\\
 
129 or 43 &= (2y+2x+43)\\
−
1 or 3&= 2y-2x+3\\
+
1 or 3 &= 2y-2x+3\\
 
\end{align}</cmath>
 
\end{align}</cmath>
−
43 & 1 yields (0,0) which is not what we want.
+
43 & 3 yields (0,0) which is not what we want.
 
129 & 1 yields (22,21) which is more interesting.
 
129 & 1 yields (22,21) which is more interesting.
  
Line 55: Line 60:
  
 
~Technodoggo
 
~Technodoggo
 +
 +
== Solution 4 ==
 +
 +
Say one factorization is <math>n(n+23).</math> The two cases for the other factorization are <math>(n+1)(n+21)</math> and <math>(n+2)(n+22).</math> We know it must be the first because of AM-GM intuition: lesser factors of a number are closer together than larger factors of a number. (We can also try both and see which works.) Thus, <math>n(n+23)=(n+1)(n+21)</math> and we find that <math>n=21,N=924</math> meaning the answer is <math>\boxed{\textbf{(C) }15}.</math>
 +
 +
~DouDragon
 +
 +
== Solution 5 ==
 +
 +
Since we are given that some pairs of divisors differ by 20 and 23 and we can let the pair be <math>(x-10)</math> and <math>(x+10)</math> as well as <math>(y-\frac{23}{2})</math> and <math>(y+\frac{23}{2})</math> . We also know the product of both the complementary divisors give the same number so <math>(x-10)(x+10)=(y-\frac{23}{2})(y+\frac{23}{2})</math> .
 +
Now we let <math>y=\frac{a}{2}</math>. Then we substitute and get <math>x^2-100=\frac{(a^2-529)}{4}</math>. Finally we multiply by 4 and get <math>4x^2-a^2=-129, a^2-4x^2=129</math>.
 +
Then we use differences of squares and get <math>a</math>+<math>2x</math>=129, <math>a</math>-<math>2x</math>=1. We finish by getting <math>a=</math>65 and <math>x=32</math>. So <math>(42)(22) = 924</math> Adding the digits, we have <math>9+2+4 = \boxed{\textbf{(C) } 15}</math>.
 +
 +
 +
~averageguy
 +
 +
 +
Nunber sense note: To avoid tedious multiplication of 2-digit numbers, observe that <math>n = (42)(22) = (6)(7)(2)(11)</math>, and <math>(6)(7)(2) = 84</math>, and the sum of the digits of <math>11</math> is <math>2</math>,  so the sum of the digits of <math>n</math> is equivalent to <math>(8+4)(2) \equiv 24 \equiv 15 \pmod 9</math>.  The only equivalent answer choice is  <math>\boxed{15}</math>. ~oinava
 +
 +
==Solution 6==
 +
 +
<math>N</math> can be written <math>N = \left( a - 10 \right) \left( a + 10 \right)</math> with a positive integer <math>a > 10</math> and <math>N = \left( \frac{2b + 1}{2} - \frac{23}{2} \right) \left( \frac{2b + 1}{2} + \frac{23}{2} \right)</math> with a positive integer <math>b > 11</math>.
 +
 +
The above equations can be reorganized as
 +
<cmath>
 +
\[
 +
\left( 2b + 1 + 2 a \right) \left( 2 b + 1 - 2 a \right)
 +
= 43 \cdot 3 .
 +
\]
 +
</cmath>
 +
 +
The only solution is <math>2b + 1 + 2a = 129</math> and <math>2b + 1 - 2a = 1</math>.
 +
Thus, <math>a = b = 32</math>.
 +
Therefore, <math>N = 924</math>.
 +
So the sum of the digits of <math>N</math> is <math>9 + 2 + 4 = \boxed{\textbf{(C)}~15}</math>.
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 +
==Solution 7==
 +
We can write <math>N</math> as <math>a(a+20)</math> or <math>b(b+23)</math> where <math>a</math> and <math>b</math> are divisors of <math>N.</math> Since <math>a(a+20) = b(b+23),</math> we know that <math>a^2 + 20a - b^2 - 23b = 0</math>, and we can view this as a quadratic in <math>a.</math>
 +
 +
Since the solution for <math>a</math> must be an integer, the discriminant for this quadratic must be a perfect square and therefore <math>20^2 - 4(-b^2 - 23b) = (2c)^2 = 400 + 4b^2 + 92b</math> so <math>b^2 + 23b -c^2 + 100 = 0.</math>
 +
 +
Since the discriminant of this quadratic in <math>b</math> must also be a perfect square we know that <math>23^2 - 4(-c^2+100) = d^2</math> which we can simplify as <math>d^2 - 4c^2 = (d-2c)(d+2c) = 129.</math> Since they are both positive integers <math>d - 2c</math> and <math>d + 2c</math> are factors of <math>129 = 3 \cdot 43</math> so <math>d - 2c = 1</math> and <math>d + 2c = 129</math> or <math>d - 2c = 3</math> and <math>d - 2c = 43.</math>
 +
 +
These systems of equations give us <math>(c,d) = (32,65)</math> and <math>(c,d) = (10,23)</math> respectively, if we plug our values for <math>c</math> into the equation for <math>b</math> we get <math>b^2 + 23b - 924 = 0</math> and <math>b^2 + 23b = 0</math> respectively. The first equation gives us <math>b = 21</math> or <math>b = -44</math> and the second gives us <math>b = 0</math> or <math>b = -23</math>, since <math>b</math> is positive we know that <math>b = 21</math> and <math>N = (21)(21 + 23) = 924</math>, therefore the sum of the digits of <math>N</math> is <math>9 + 2 + 4 = \boxed{\textbf{(C) 15}}.</math>
 +
 +
~SailS
 +
 +
==Solution 8 (Trial and Error)==
 +
Consider the numbers of the form <math>a(a+20)</math>. Since <math>b(b+23)</math> is always even, <math>a</math> is even. Thus, for <math>a \ge 2</math>, we calculate <math>a(a+20)</math> for even values of <math>a</math>. Then, we check if it can also be represented as a product of numbers that differ by <math>23</math>. Checking, we see that <math>22 \cdot 42 = 21 \cdot 44 = 924</math> works. Thus, the answer is <math>9 + 2 + 4 = \boxed{\textbf{(C) 15}}</math>
 +
 +
~andliu766
 +
 +
==Solution 9==
 +
 +
<math>n(n+20)=m(m+23) \Longrightarrow n^2+20n=m^2+23m \Longrightarrow n^2-m^2+20n-20m=3m</math>. Factoring, <math>(n+m)(n-m)+20(n-m)=3m \Longrightarrow (n-m)(n+m+20)=3m</math>. Let <math>n-m=a>0</math> because clearly <math>n>m</math>. Then. <math>a(2m+20+a)=3m</math>. Note that since <math>20+a>0</math>, if <math>a\geq2</math>, then the equation is <math>4m+a(20+a)>3m</math>, so <math>a-1</math>. Plugging this back, we get <math>2m+21=3m \Longrightarrow m=21</math> and <math>n=22</math>. Now we find <math>N</math> as <math>22*42=924</math> so the answer is <math>15</math>.
 +
 +
-Magnetoninja
 +
 +
==Video Solution by Little Fermat==
 +
https://youtu.be/h2Pf2hvF1wE?si=qPO3xUAoaBPvvkd2&t=5118
 +
~little-fermat
 +
==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/GP-DYudh5qU?si=gxvKNnXX1gjgdkvP&t=8645
 +
 +
~Math-X
 +
 +
==Video Solution ⚡️ 3 min solution ⚡️ ==
 +
 +
https://youtu.be/fuH_b6AieCQ
 +
 +
<i> ~Education, the Study of Everything </i>
  
 
== Video Solution 1 by OmegaLearn ==
 
== Video Solution 1 by OmegaLearn ==
 
https://youtu.be/D_T24PrVk18
 
https://youtu.be/D_T24PrVk18
 +
 +
==Video Solution by epicbird08==
 +
https://youtu.be/HrZ3fia7g2A
 +
 +
~EpicBird08
 +
 +
==Video Solution==
 +
 +
https://youtu.be/J9VAVT22L40
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2023|ab=A|num-b=22|num-a=24}}
 
{{AMC10 box|year=2023|ab=A|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 06:37, 5 November 2024

Problem

If the positive integer $n$ has positive integer divisors $a$ and $b$ with $n = ab$, then $a$ and $b$ are said to be $\textit{complementary}$ divisors of $n$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. What is the sum of the digits of $N$?

$\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

Solution 1

Consider positive integers $a, b$ with a difference of $20$. Suppose $b = a-20$. Then, we have $(a)(a-20) = n$. If there is another pair of two integers that multiply to $n$ but have a difference of 23, one integer must be greater than $a$, and the other must be smaller than $a-20$. We can create two cases and set both equal.

We have $(a)(a-20) = (a+1)(a-22) \text{or} (a+2)(a-21).$ Note that if we go further to $(a+3)(a-20)$ and beyond, that would violate the condition that one of the two integers must be smaller than $a-20.$

Starting with the first case, we have $a^2-20a = a^2-21a-22$,or $0=-a-22$, which gives $a=-22$, which is not possible. The other case is $a^2-20a = a^2-19a-42$, so $a=42$. Thus, our product is $(42)(22) = (44)(21)$, so $c = 924$. Adding the digits, we have $9+2+4 = \boxed{\textbf{(C) } 15}$.

-Sepehr2010, mathboy282, the_eaglercraft_grinder

Solution 2

We have 4 integers in our problem. Let's call the smallest of them $a$. $a(a+23) =$ either $(a+1)(a+21)$ or $(a+2)(a+22)$. So, we have the following:

$a^2 + 23a = a^2 + 22a +21$ or

$a^2+23a = a^2 + 24a +44$.

The second equation has negative solutions, so we discard it. The first equation has $a = 21$, and so $a + 23 = 44$. If we check $(a+1)(a+21)$ we get $22 \cdot 42 = 21 \cdot 44$. $44$ is $2$ times $22$, and $42$ is $2$ times $21$, so our solution checks out. Multiplying $21$ by $44$, we get $924$ => $9 + 2 + 4 = \boxed{\textbf{(C) 15}}$.

~Arcticturn

Solution 3

From the problems, it follows that

\begin{align*} x(x+20)&=y(y+23) = N\\ x^2+20x&=y^2+23y\\ 4x^2+4\cdot20x &= 4y^2+4\cdot23y\\ 4x^2+4\cdot20x+20^2-20^2 &= 4y^2+4\cdot23y+23^2-23^2\\ (2x+20)^2-20^2 &= (2y+23)^2-23^2\\ 23^2-20^2 &= (2y+23)^2-(2x+20)^2\\ (23+20)(23-20) &= (2y+23+2x+20)(2y+23-2x-20)\\ 43\cdot 3 &= (2y+2x+43)(2y-2x+3)\\ 129\cdot 1 &= (2y+2x+43)(2y-2x+3)\\ \end{align*} Since both $(2y+2x+43)$ and $(2y-2x+3)$ must be integer, we get two equations. \begin{align} 129 or 43 &= (2y+2x+43)\\ 1 or 3 &= 2y-2x+3\\ \end{align} 43 & 3 yields (0,0) which is not what we want. 129 & 1 yields (22,21) which is more interesting.

Simplifying the equations, we get: \begin{align*} x+y &= 43\\ x-y &= 1\\ x=22&, y=21\\ N &= (22)(22+20) = 924. \end{align*}

So, the answer is $\boxed{\textbf{(C) 15}}$.


~Technodoggo

Solution 4

Say one factorization is $n(n+23).$ The two cases for the other factorization are $(n+1)(n+21)$ and $(n+2)(n+22).$ We know it must be the first because of AM-GM intuition: lesser factors of a number are closer together than larger factors of a number. (We can also try both and see which works.) Thus, $n(n+23)=(n+1)(n+21)$ and we find that $n=21,N=924$ meaning the answer is $\boxed{\textbf{(C) }15}.$

~DouDragon

Solution 5

Since we are given that some pairs of divisors differ by 20 and 23 and we can let the pair be $(x-10)$ and $(x+10)$ as well as $(y-\frac{23}{2})$ and $(y+\frac{23}{2})$ . We also know the product of both the complementary divisors give the same number so $(x-10)(x+10)=(y-\frac{23}{2})(y+\frac{23}{2})$ . Now we let $y=\frac{a}{2}$. Then we substitute and get $x^2-100=\frac{(a^2-529)}{4}$. Finally we multiply by 4 and get $4x^2-a^2=-129, a^2-4x^2=129$. Then we use differences of squares and get $a$+$2x$=129, $a$-$2x$=1. We finish by getting $a=$65 and $x=32$. So $(42)(22) = 924$ Adding the digits, we have $9+2+4 = \boxed{\textbf{(C) } 15}$.


~averageguy


Nunber sense note: To avoid tedious multiplication of 2-digit numbers, observe that $n = (42)(22) = (6)(7)(2)(11)$, and $(6)(7)(2) = 84$, and the sum of the digits of $11$ is $2$, so the sum of the digits of $n$ is equivalent to $(8+4)(2) \equiv 24 \equiv 15 \pmod 9$. The only equivalent answer choice is $\boxed{15}$. ~oinava

Solution 6

$N$ can be written $N = \left( a - 10 \right) \left( a + 10 \right)$ with a positive integer $a > 10$ and $N = \left( \frac{2b + 1}{2} - \frac{23}{2} \right) \left( \frac{2b + 1}{2} + \frac{23}{2} \right)$ with a positive integer $b > 11$.

The above equations can be reorganized as \[ \left( 2b + 1 + 2 a \right) \left( 2 b + 1 - 2 a \right) = 43 \cdot 3 . \]

The only solution is $2b + 1 + 2a = 129$ and $2b + 1 - 2a = 1$. Thus, $a = b = 32$. Therefore, $N = 924$. So the sum of the digits of $N$ is $9 + 2 + 4 = \boxed{\textbf{(C)}~15}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 7

We can write $N$ as $a(a+20)$ or $b(b+23)$ where $a$ and $b$ are divisors of $N.$ Since $a(a+20) = b(b+23),$ we know that $a^2 + 20a - b^2 - 23b = 0$, and we can view this as a quadratic in $a.$

Since the solution for $a$ must be an integer, the discriminant for this quadratic must be a perfect square and therefore $20^2 - 4(-b^2 - 23b) = (2c)^2 = 400 + 4b^2 + 92b$ so $b^2 + 23b -c^2 + 100 = 0.$

Since the discriminant of this quadratic in $b$ must also be a perfect square we know that $23^2 - 4(-c^2+100) = d^2$ which we can simplify as $d^2 - 4c^2 = (d-2c)(d+2c) = 129.$ Since they are both positive integers $d - 2c$ and $d + 2c$ are factors of $129 = 3 \cdot 43$ so $d - 2c = 1$ and $d + 2c = 129$ or $d - 2c = 3$ and $d - 2c = 43.$

These systems of equations give us $(c,d) = (32,65)$ and $(c,d) = (10,23)$ respectively, if we plug our values for $c$ into the equation for $b$ we get $b^2 + 23b - 924 = 0$ and $b^2 + 23b = 0$ respectively. The first equation gives us $b = 21$ or $b = -44$ and the second gives us $b = 0$ or $b = -23$, since $b$ is positive we know that $b = 21$ and $N = (21)(21 + 23) = 924$, therefore the sum of the digits of $N$ is $9 + 2 + 4 = \boxed{\textbf{(C) 15}}.$

~SailS

Solution 8 (Trial and Error)

Consider the numbers of the form $a(a+20)$. Since $b(b+23)$ is always even, $a$ is even. Thus, for $a \ge 2$, we calculate $a(a+20)$ for even values of $a$. Then, we check if it can also be represented as a product of numbers that differ by $23$. Checking, we see that $22 \cdot 42 = 21 \cdot 44 = 924$ works. Thus, the answer is $9 + 2 + 4 = \boxed{\textbf{(C) 15}}$

~andliu766

Solution 9

$n(n+20)=m(m+23) \Longrightarrow n^2+20n=m^2+23m \Longrightarrow n^2-m^2+20n-20m=3m$. Factoring, $(n+m)(n-m)+20(n-m)=3m \Longrightarrow (n-m)(n+m+20)=3m$. Let $n-m=a>0$ because clearly $n>m$. Then. $a(2m+20+a)=3m$. Note that since $20+a>0$, if $a\geq2$, then the equation is $4m+a(20+a)>3m$, so $a-1$. Plugging this back, we get $2m+21=3m \Longrightarrow m=21$ and $n=22$. Now we find $N$ as $22*42=924$ so the answer is $15$.

-Magnetoninja

Video Solution by Little Fermat

https://youtu.be/h2Pf2hvF1wE?si=qPO3xUAoaBPvvkd2&t=5118 ~little-fermat

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/GP-DYudh5qU?si=gxvKNnXX1gjgdkvP&t=8645

~Math-X

Video Solution ⚡️ 3 min solution ⚡️

https://youtu.be/fuH_b6AieCQ

~Education, the Study of Everything

Video Solution 1 by OmegaLearn

https://youtu.be/D_T24PrVk18

Video Solution by epicbird08

https://youtu.be/HrZ3fia7g2A

~EpicBird08

Video Solution

https://youtu.be/J9VAVT22L40

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

See Also

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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