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Difference between revisions of "1962 AHSME Problems/Problem 38"

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<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 17 </math>
 
<math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 17 </math>
  
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==Solution==
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==Solution 1==
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{{solution}}
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Let <math>a^2</math> <math>=</math> original population count, <math>b^2+1</math> <math>=</math> the second population count, and <math>c^2</math> <math>=</math> the third population count
 +
We first see that <math>a^2 + 100 = b^2 + 1</math> or <math>99</math> <math>=</math> <math>b^2-a^2</math>.
 +
We then factor the right side getting <math>99</math> <math>=</math> <math>(b-a)(b+a)</math>.
 +
Since we can only have an nonnegative integral population, clearly <math>b+a</math> <math>></math> <math>b-a</math> and both factor <math>99</math>.
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We factor <math>99</math> into <math>3^2 \cdot 11</math> <math>=</math> <math>(b-a)(b+a)</math>
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There are a few cases to look at:
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<math>1)</math> <math>b+a</math> <math>=</math> <math>11</math> and <math>b-a</math> <math>=</math> <math>9</math>.
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Adding the two equations we get <math>2b</math> <math>=</math> <math>20</math> or <math>b</math> <math>=</math> <math>10</math>, which means <math>a</math> <math>=</math> <math>1</math>.
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But looking at the restriction that the second population + <math>100</math> <math>=</math> third population...
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<math>10^2</math> <math>+</math> <math>1</math> <math>+</math> <math>100</math> <math>=</math> <math>201</math> <math>\neq</math> a perfect square.
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<math>2)</math> <math>b+a</math> <math>=</math> <math>33</math> and <math>b-a</math> <math>=</math> <math>3</math>.
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Adding the two equations we get <math>2b</math> <math>=</math> <math>36</math> or <math>b</math> <math>=</math> <math>18</math>, which means <math>a</math> <math>=</math> <math>15</math>.
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Looking at the same restriction, we get <math>18^2</math> + <math>1</math> + <math>100</math> <math>=</math> <math>324</math> + <math>101</math> <math>=</math> <math>425</math>, which is NOT a perfect square.
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Finally, <math>b+a</math> <math>=</math> <math>99</math> and <math>b-a</math> <math>=</math> <math>1</math>.
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<math>2b</math> <math>=</math> <math>100</math> or <math>b</math> <math>=</math> <math>50</math>, which means <math>a</math> <math>=</math> <math>49</math>.
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Looking at the same restriction, we get <math>50^2</math> + <math>1</math> + <math>100</math> <math>=</math> <math>2500</math> + <math>101</math> <math>=</math> <math>2601</math> <math>=</math> <math>51^2</math>. Thus we find that the original population is <math>a^2</math> <math>=</math> <math>49^2</math> <math>=</math> <math>7^4</math>. Or <math>a^2</math> is a multiple of <math>\boxed{ (B) 7}</math>
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==Solution 2==
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Let <math>P</math> <math>=</math> original population. Translating the word problem into a system of equations, we got:
 +
<cmath>
 +
\begin{align}
 +
P &= x^2 \\
 +
P + 100 &= y^2 + 1 \\
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P + 200 &= z^2
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\end{align}
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</cmath>
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for some positive integers <math>x</math>, <math>y</math> and <math>z</math>.
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Now, by subtracting <math>(2)</math> from <math>(3)</math> (i.e. <math>(3) - (2)</math>), we got:
 +
<cmath>
 +
\begin{align*}
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100 &= z^2 - y^2 - 1 \\
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101 &= z^2 - y^2 \\
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101 &= (z - y)(z + y)
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\end{align*}
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</cmath>
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Since y and z are both positive integers and 101 is a prime, by factoring, the only working solution for us is <math>y = 50</math> and <math>z = 51</math>.
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Plugging that back to <math>(2)</math> and simplify, we got <math>P = 2401 = (49)^2 = x^2</math>, a multiple of <math>7</math>.
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Therefore, the answer is <math>\boxed{(B) 7}</math>. -nullptr07

Latest revision as of 15:52, 28 June 2023

Problem

The population of Nosuch Junction at one time was a perfect square. Later, with an increase of $100$, the population was one more than a perfect square. Now, with an additional increase of $100$, the population is again a perfect square.


The original population is a multiple of:

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 17$

Solution 1

Let $a^2$ $=$ original population count, $b^2+1$ $=$ the second population count, and $c^2$ $=$ the third population count We first see that $a^2 + 100 = b^2 + 1$ or $99$ $=$ $b^2-a^2$. We then factor the right side getting $99$ $=$ $(b-a)(b+a)$. Since we can only have an nonnegative integral population, clearly $b+a$ $>$ $b-a$ and both factor $99$. We factor $99$ into $3^2 \cdot 11$ $=$ $(b-a)(b+a)$ There are a few cases to look at: $1)$ $b+a$ $=$ $11$ and $b-a$ $=$ $9$. Adding the two equations we get $2b$ $=$ $20$ or $b$ $=$ $10$, which means $a$ $=$ $1$. But looking at the restriction that the second population + $100$ $=$ third population... $10^2$ $+$ $1$ $+$ $100$ $=$ $201$ $\neq$ a perfect square.

$2)$ $b+a$ $=$ $33$ and $b-a$ $=$ $3$. Adding the two equations we get $2b$ $=$ $36$ or $b$ $=$ $18$, which means $a$ $=$ $15$. Looking at the same restriction, we get $18^2$ + $1$ + $100$ $=$ $324$ + $101$ $=$ $425$, which is NOT a perfect square.

Finally, $b+a$ $=$ $99$ and $b-a$ $=$ $1$. $2b$ $=$ $100$ or $b$ $=$ $50$, which means $a$ $=$ $49$. Looking at the same restriction, we get $50^2$ + $1$ + $100$ $=$ $2500$ + $101$ $=$ $2601$ $=$ $51^2$. Thus we find that the original population is $a^2$ $=$ $49^2$ $=$ $7^4$. Or $a^2$ is a multiple of $\boxed{ (B) 7}$

Solution 2

Let $P$ $=$ original population. Translating the word problem into a system of equations, we got: \begin{align} P &= x^2 \\ P + 100 &= y^2 + 1 \\ P + 200 &= z^2 \end{align} for some positive integers $x$, $y$ and $z$. Now, by subtracting $(2)$ from $(3)$ (i.e. $(3) - (2)$), we got: \begin{align*} 100 &= z^2 - y^2 - 1 \\ 101 &= z^2 - y^2 \\ 101 &= (z - y)(z + y) \end{align*} Since y and z are both positive integers and 101 is a prime, by factoring, the only working solution for us is $y = 50$ and $z = 51$. Plugging that back to $(2)$ and simplify, we got $P = 2401 = (49)^2 = x^2$, a multiple of $7$. Therefore, the answer is $\boxed{(B) 7}$. -nullptr07

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