Difference between revisions of "2012 AIME I Problems/Problem 10"

Latest revision as of 17:08, 7 September 2024

Problem

Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x-256}{1000}$ , where $x$ is in $\mathcal{S}$ . In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$ .

Solution 1

It is apparent that for a perfect square $s^2$ to satisfy the constraints, we must have $s^2 - 256 = 1000n$ or $(s+16)(s-16) = 1000n.$ Now in order for $(s+16)(s-16)$ to be a multiple of $1000,$ at least one of $s+16$ and $s-16$ must be a multiple of $5,$ and since $s+16 \not\equiv s-16 \pmod{5},$ one term must have all the factors of $5$ and thus must be a multiple of $125.$ Furthermore, each of $s+16$ and $s-16$ must have at least two factors of $2,$ since otherwise $(s+16)(s-16)$ could not possibly be divisible by $8.$ So therefore the conditions are satisfied if either $s+16$ or $s-16$ is divisible by $500,$ or equivalently if $s = 500n \pm 16.$ Counting up from $n=0$ to $n=5,$ we see that the tenth value of $s$ is $500 \cdot 5 - 16 = 2484$ and thus the corresponding element in $\mathcal{T}$ is $\frac{2484^2 - 256}{1000} = 6170 \rightarrow \boxed{170.}$

Solution 2

Notice that is $16^2=256$ , $1016^2$ ends in $256$ . In general, if $x^2$ ends in $256$ , $(x+1000)^2=x^2+2000x+1000000$ ends in 256 because $1000|2000x$ and $1000|2000000$ . It is clear that we want all numbers whose squares end in $256$ that are less than $1000$ .

Firstly, we know the number has to end in a $4$ or a $6$ . Let's look at the ones ending in $6$ .

Assume that the second digit of the three digit number is $0$ . We find that the last $3$ digits of $\overline{a06}^2$ is in the form $12a \cdot 100 + 3 \cdot 10 + 6$ . However, the last two digits need to be a $56$ . Thus, similarly trying all numbers from $0$ to $10$ , we see that only 1 for the middle digit works. Carrying out the multiplication, we see that the last $3$ digits of $\overline{a16}^2$ is in the form $(12a + 2) \cdot 100 + 5 \cdot 10 + 6$ . We want $(12a + 2)\pmod{10}$ to be equal to $2$ . Thus, we see that a is $0$ or $5$ . Thus, the numbers that work in this case are $016$ and $516$ .

Next, let's look at the ones ending in $4$ . Carrying out a similar technique as above, we see that the last $3$ digits of $\overline{a84}^2$ is in the form $((8a+10) \cdot 100+ 5 \cdot 10 + 6$ . We want $(8a + 10)\pmod{10}$ to be equal to $2$ . We see that only $4$ and $9$ work. Thus, we see that only $484$ and $984$ work.

We order these numbers to get $16$ , $484$ , $516$ , $984$ ... We want the $10th$ number in order which is $2484^2 = 6\boxed{170}256$ .

Solution 3

The condition implies $x^2\equiv 256 \pmod{1000}$ . Rearranging and factoring, $(x-16)(x+16)\equiv 0\pmod {1000}.$ This can be expressed with the system of congruences $\begin{cases} (x-16)(x+16)\equiv 0\pmod{125} \\ (x-16)(x+16)\equiv 0\pmod{8} \end{cases}$ Observe that $x\equiv {109} \pmod {125}$ or $x\equiv{16}\pmod {125}$ . Similarly, it can be seen that $x\equiv{0}\pmod{8}$ or $x\equiv{4}\pmod{8}$ . By CRT, there are four solutions to this modulo $1000$ (one for each case e.g. $x\equiv{109}$ and $x\equiv{0}$ or $x\equiv{125}$ and $x\equiv{4}$ . These solutions are (working modulo $1000$ ) $\begin{cases} x=16 \\ x=484 \\ x= 516 \\ x=984 \end{cases}$ The tenth solution is $x=2484,$ which gives an answer of $\boxed{170}$ .

Solution 4

An element in S can be represented by $y^2 = 1000a + 256$ , where $y^2$ is the element in S. Since the right hand side must be even, we let $y = 2y_1$ and substitute to get $y_1^2 = 250a + 64$ . However, the right hand side is still even, so we make the substitution $y_1 = 2y_2$ to get $y_2^2 = 125a/2 + 16$ . Because both sides must be an integer, we know that $a = 2a_1$ for some integer $a_1$ . Our equation then becomes $y_2^2 = 125a_1 + 16$ , and we can simplify no further.

Rearranging terms, we get $y_2^2 - 16 = 125a_1$ , whence difference of squares gives $(y_2 + 4)(y_2 - 4) = 125a_1$ . Note that this equation tells us that one of $y_2 + 4$ and $y_2 - 4$ contains a nonnegative multiple of $125$ . Hence, listing out the smallest possible values of $y_2$ , we have $y_2 = 4, 121, 129, \cdots, 621$ . The tenth term is $621$ , whence $y = 4y_2 = 2484$ . The desired result can then be calculated to be $\boxed{170}$ . - Spacesam

Solution 5 (Similar to Solution 4)

From the conditions, we can let every element in $\mathcal{S}$ be written as $y^2=1000x+256$ , where $x$ and $y$ are integers. Since there are no restrictions on $y$ , we let $y_1$ be equal to $y+16$ ( $y-16$ works as well). Then the $256$ cancels out and we're left with $y_1^2+32y_1=1000x$ which can be factored as $y_1(y_1+32)=1000x$ Since the RHS is even, $y_1$ must be even, so we let $y_1=2y_2$ , to get $y_2(y_2+16)=250x$ Again, because the RHS is even, the LHS must be even too, so substituting $y_3=\frac{1}{2}y_2$ we have $y_3(y_3+8)=125\cdot\frac{1}{2}x$ Since the LHS is an integer, the RHS must thus be an integer, so substituting $x=2x_1$ we get $y_3(y_3+8)=125x_1$ Then we can do casework on the values of $y_3$ , as only one of $y_3$ and $y_3+8$ can be multiples of $125$

Case 1: $125|y_3$

Since we're trying to find the values of $x_1$ , we can let $y_4=\frac{1}{125}y_3$ , to get $x_1=y_4(125y_4+8)$ or $x=2y_4(125y_4+8)$

Case 2: $125|y_3+8$

Similar to Case 1, only the equation is $x=2y_4(125y_4-8)$

In whole, the values of $x$ (i.e. the elements in $\mathcal{T}$ ) are of the form $x=2k(125k\pm8)$ where $k$ is any integer. It can easily be seen that if $k<0$ , then $x$ is negative, thus $k\geq0$ . Also, note that when $k=0$ , there is only one value, because one of the factors is $0$ ( $k$ ). Thus the $10^{th}$ smallest number in the set $\mathcal{T}$ is when the $\pm$ sign takes the minus side and $k=5$ , giving $6170$ , so the answer is $\boxed{170}$

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012aimei/349

~ dolphin7

Video Solution

https://www.youtube.com/watch?v=caCELHibbIE&list=PLOaAlyCEsUTbA2v1gRyEAB_gxk7NiWG3v&index=6&t=0s

~Shreyas S

@@ Line 1: / Line 1: @@
-==Problem 10==
+==Problem==
 Let <math>\mathcal{S}</math> be the set of all perfect squares whose rightmost three digits in base <math>10</math> are <math>256</math>. Let <math>\mathcal{T}</math> be the set of all numbers of the form <math>\frac{x-256}{1000}</math>, where <math>x</math> is in <math>\mathcal{S}</math>. In other words, <math>\mathcal{T}</math> is the set of numbers that result when the last three digits of each number in <math>\mathcal{S}</math> are truncated. Find the remainder when the tenth smallest element of <math>\mathcal{T}</math> is divided by <math>1000</math>.
 == Solution 1==
-It is apparent that for a perfect square <math>s^2</math> to satisfy the constraints, we must have <math>s^2 - 256 = 1000n</math> or <math>(s+16)(s-16) = 1000n.</math> Now in order for <math>(s+16)(s-16)</math> to be a multiple of <math>1000,</math> at least one of <math>s+16</math> and <math>s-16</math> must be a multiple of <math>5,</math> and since <math>s+16</math> and <math>s-16</math> are in different residue classes mod <math>5,</math> one term must have all the factors of <math>5</math> and thus must be a multiple of <math>125.</math> Furthermore, each of <math>s+16</math> and <math>s-16</math> must have at least two factors of <math>2,</math> since otherwise <math>(s+16)(s-16)</math> could not possibly be divisible by <math>8.</math> So therefore the conditions are satisfied if either <math>s+16</math> or <math>s-16</math> is divisible by <math>500,</math> or equivalently if <math>s = 500n \pm 16.</math> Counting up from <math>n=0</math> to <math>n=5,</math> we see that the tenth value of <math>s</math> is <math>500 \cdot 5 - 16 = 2484</math> and thus the corresponding element in <math>\mathcal{T}</math> is <math>\frac{2484^2 - 256}{1000} = 6170 \rightarrow \boxed{170.}</math>
+It is apparent that for a perfect square <math>s^2</math> to satisfy the constraints, we must have <math>s^2 - 256 = 1000n</math> or <math>(s+16)(s-16) = 1000n.</math> Now in order for <math>(s+16)(s-16)</math> to be a multiple of <math>1000,</math> at least one of <math>s+16</math> and <math>s-16</math> must be a multiple of <math>5,</math> and since <math>s+16 \not\equiv s-16 \pmod{5},</math> one term must have all the factors of <math>5</math> and thus must be a multiple of <math>125.</math> Furthermore, each of <math>s+16</math> and <math>s-16</math> must have at least two factors of <math>2,</math> since otherwise <math>(s+16)(s-16)</math> could not possibly be divisible by <math>8.</math> So therefore the conditions are satisfied if either <math>s+16</math> or <math>s-16</math> is divisible by <math>500,</math> or equivalently if <math>s = 500n \pm 16.</math> Counting up from <math>n=0</math> to <math>n=5,</math> we see that the tenth value of <math>s</math> is <math>500 \cdot 5 - 16 = 2484</math> and thus the corresponding element in <math>\mathcal{T}</math> is <math>\frac{2484^2 - 256}{1000} = 6170 \rightarrow \boxed{170.}</math>
 == Solution 2==
-Notice that is <math>16^2=256</math>, <math>1016^2</math> ends in <math>256</math>. In general, if <math>x^2</math> ends in <math>256</math>, <math>(x+1000)^2=x^2+2000x+1000000</math> ends in 256 because <math>2000x >1000</math> and <math>2000000 > 1000</math>. It is clear that we want all numbers whose squares end in <math>256</math> that are less than <math>1000</math>.
+Notice that is <math>16^2=256</math>, <math>1016^2</math> ends in <math>256</math>. In general, if <math>x^2</math> ends in <math>256</math>, <math>(x+1000)^2=x^2+2000x+1000000</math> ends in 256 because <math>1000|2000x</math> and <math>1000|2000000</math>. It is clear that we want all numbers whose squares end in <math>256</math> that are less than <math>1000</math>.
 Firstly, we know the number has to end in a <math>4</math> or a <math>6</math>. Let's look at the ones ending in <math>6</math>.
-Assume that the second digit of the three digit number is <math>0</math>. We find that the last <math>3</math> digits of <math>\overline{a06}^2</math> is in the form <math>12a \cdot 100 + 3 \cdot 10 + 6</math>. However, the last two digits need to be a <math>56</math>. Thus, similarly trying all numbers from <math>0</math> to <math>10</math>, we see that only 1 for the middle digit works. Carrying out the multiplication, we see that the last <math>3</math> digits of <math>\overline{a06}^2</math> is in the form <math>(12a + 2) \cdot 100 + 5 \cdot 10 + 6</math>. We want <math>(12a + 2)\pmod{10}</math> to be equal to <math>2</math>. Thus, we see that a is <math>0</math> or <math>5</math>. Thus, the numbers that work in this case are <math>016</math> and <math>516</math>.
+Assume that the second digit of the three digit number is <math>0</math>. We find that the last <math>3</math> digits of <math>\overline{a06}^2</math> is in the form <math>12a \cdot 100 + 3 \cdot 10 + 6</math>. However, the last two digits need to be a <math>56</math>. Thus, similarly trying all numbers from <math>0</math> to <math>10</math>, we see that only 1 for the middle digit works. Carrying out the multiplication, we see that the last <math>3</math> digits of <math>\overline{a16}^2</math> is in the form <math>(12a + 2) \cdot 100 + 5 \cdot 10 + 6</math>. We want <math>(12a + 2)\pmod{10}</math> to be equal to <math>2</math>. Thus, we see that a is <math>0</math> or <math>5</math>. Thus, the numbers that work in this case are <math>016</math> and <math>516</math>.
 Next, let's look at the ones ending in <math>4</math>. Carrying out a similar technique as above, we see that the last <math>3</math> digits of <math>\overline{a84}^2</math> is in the form <math>((8a+10) \cdot 100+ 5 \cdot 10 + 6</math>. We want <math>(8a + 10)\pmod{10}</math> to be equal to <math>2</math>. We see that only <math>4</math> and <math>9</math> work. Thus, we see that only <math>484</math> and <math>984</math> work.
-We order these numbers to get  <math>16</math>, <math>484</math>, <math>516</math>, <math>984</math>... We want the <math>10th</math> number in order which is <math>2484^2 = 6170256</math>.  <math>\boxed{170}</math>
+We order these numbers to get  <math>16</math>, <math>484</math>, <math>516</math>, <math>984</math>... We want the <math>10th</math> number in order which is <math>2484^2 = 6\boxed{170}256</math>.
 == Solution 3==
@@ Line 38: / Line 38: @@
 Rearranging terms, we get <math>y_2^2 - 16 = 125a_1</math>, whence difference of squares gives <math>(y_2 + 4)(y_2 - 4) = 125a_1</math>. Note that this equation tells us that one of <math>y_2 + 4</math> and <math>y_2 - 4</math> contains a nonnegative multiple of <math>125</math>. Hence, listing out the smallest possible values of <math>y_2</math>, we have <math>y_2 = 4, 121, 129, \cdots, 621</math>. The tenth term is <math>621</math>, whence <math>y = 4y_2 = 2484</math>. The desired result can then be calculated to be <math>\boxed{170}</math>. - Spacesam
-=== Video Solution by Richard Rusczyk ===
+== Solution 5 (Similar to Solution 4) ==
+From the conditions, we can let every element in <math>\mathcal{S}</math> be written as <math>y^2=1000x+256</math>, where <math>x</math> and <math>y</math> are integers. Since there are no restrictions on <math>y</math>, we let <math>y_1</math> be equal to <math>y+16</math> (<math>y-16</math> works as well). Then the <math>256</math> cancels out and we're left with <cmath>y_1^2+32y_1=1000x</cmath> which can be factored as <cmath>y_1(y_1+32)=1000x</cmath> Since the RHS is even, <math>y_1</math> must be even, so we let <math>y_1=2y_2</math>, to get <cmath>y_2(y_2+16)=250x</cmath> Again, because the RHS is even, the LHS must be even too, so substituting <math>y_3=\frac{1}{2}y_2</math> we have <cmath>y_3(y_3+8)=125\cdot\frac{1}{2}x</cmath> Since the LHS is an integer, the RHS must thus be an integer, so substituting <math>x=2x_1</math> we get <cmath>y_3(y_3+8)=125x_1</cmath> Then we can do casework on the values of <math>y_3</math>, as only one of <math>y_3</math> and <math>y_3+8</math> can be multiples of <math>125</math>
+'''Case 1''': <math>125|y_3</math>
+Since we're trying to find the values of <math>x_1</math>, we can let <math>y_4=\frac{1}{125}y_3</math>, to get <cmath>x_1=y_4(125y_4+8)</cmath> or <cmath>x=2y_4(125y_4+8)</cmath>
+'''Case 2''': <math>125|y_3+8</math>
+Similar to Case 1, only the equation is <cmath>x=2y_4(125y_4-8)</cmath>
+In whole, the values of <math>x</math> (i.e. the elements in <math>\mathcal{T}</math>) are of the form <cmath>x=2k(125k\pm8)</cmath> where <math>k</math> is any integer. It can easily be seen that if <math>k<0</math>, then <math>x</math> is negative, thus <math>k\geq0</math>. Also, note that when <math>k=0</math>, there is only one value, because one of the factors is <math>0</math> (<math>k</math>). Thus the <math>10^{th}</math> smallest number in the set <math>\mathcal{T}</math> is when the <math>\pm</math> sign takes the minus side and <math>k=5</math>, giving <math>6170</math>, so the answer is <math>\boxed{170}</math>
+== Video Solution by Richard Rusczyk ==
 https://artofproblemsolving.com/videos/amc/2012aimei/349
 ~ dolphin7
+==Video Solution==
+https://www.youtube.com/watch?v=caCELHibbIE&list=PLOaAlyCEsUTbA2v1gRyEAB_gxk7NiWG3v&index=6&t=0s
+~Shreyas S
 == See also ==
 {{AIME box|year=2012|n=I|num-b=9|num-a=11}}
 {{MAA Notice}}

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