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Difference between revisions of "2001 AMC 12 Problems/Problem 21"

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<cmath>
 
<cmath>
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\begin{align*}
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\begin{array}{rl}
 
ab + a + b & = 524
 
ab + a + b & = 524
 
\\  
 
\\  
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\\  
 
\\  
 
cd + c + d & = 104
 
cd + c + d & = 104
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\end{align*}
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\end{array}
 
</cmath>
 
</cmath>
  
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</math>
 
</math>
  
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== Solution 1 ==
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== Solution ==
  
 
Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
 
Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:
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Clearly <math>7^2</math> divides <math>fg</math>. On the other hand, <math>7^2</math> can not divide <math>f</math>, as it then would divide <math>ef</math>. Similarly, <math>7^2</math> can not divide <math>g</math>. Hence <math>7</math> divides both <math>f</math> and <math>g</math>. This leaves us with only two cases: <math>(f,g)=(7,21)</math> and <math>(f,g)=(21,7)</math>.
 
Clearly <math>7^2</math> divides <math>fg</math>. On the other hand, <math>7^2</math> can not divide <math>f</math>, as it then would divide <math>ef</math>. Similarly, <math>7^2</math> can not divide <math>g</math>. Hence <math>7</math> divides both <math>f</math> and <math>g</math>. This leaves us with only two cases: <math>(f,g)=(7,21)</math> and <math>(f,g)=(21,7)</math>.
  
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The first case solves to <math>(e,f,g,h)=(75,7,21,5)</math>, which gives us <math>(a,b,c,d)=(74,6,20,4)</math>, but then <math>abcd \not= 8!</math>. (We do not need to multiply, it is enough to note e.g. that the left hand side is not divisible by <math>7</math>.) Also, a - d equals <math>70</math> in this case, which is way too large to fit the answer choices.
+
The first case solves to <math>(e,f,g,h)=(75,7,21,5)</math>, which gives us <math>(a,b,c,d)=(74,6,20,4)</math>, but then <math>abcd \not= 8!</math>. We do not need to multiply, it is enough to note e.g. that the left hand side is not divisible by <math>7</math>. (Also, a - d equals <math>70</math> in this case, which is way too large to fit the answer choices.)
  
 
The second case solves to <math>(e,f,g,h)=(25,21,7,15)</math>, which gives us a valid quadruple <math>(a,b,c,d)=(24,20,6,14)</math>, and we have <math>a-d=24-14 =\boxed{10}</math>.
 
The second case solves to <math>(e,f,g,h)=(25,21,7,15)</math>, which gives us a valid quadruple <math>(a,b,c,d)=(24,20,6,14)</math>, and we have <math>a-d=24-14 =\boxed{10}</math>.
  
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== Solution 2 ==
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== Video Solution ==
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+
https://youtu.be/zlzievV5e-U
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As above, we can write the equations as follows:
 
−
 
 
−
<cmath>
 
−
\begin{align*}
 
−
(a+1)(b+1) & = 525
 
−
\\
 
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(b+1)(c+1) & = 147
 
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\\
 
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(c+1)(d+1) & = 105
 
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\end{align*}
 
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</cmath>
 
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Looking at the first two equations, we know that <math>a+1</math> but not <math>b+1</math> is a multiple of 5, and looking at the last two equations, we know that <math>(d+1)</math> but not <math>(c+1)</math> must be a multiple of 5 (since if <math>b+1</math> or <math>c+1</math> was a multiple of 5, then <math>(b+1)(c+1)</math> would also be a multiple of 5).
 
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Thus, <math>a \equiv d \equiv 0 \hspace{1 mm} \text{(mod 5)}</math>, and <math>a - d \equiv 0 \hspace{1 mm}  \text{(mod 5)}</math>. The only answer choice where this is true is <math>\boxed{\text{D) 10}}</math>.
 
  
 
== See Also ==
 
== See Also ==

Latest revision as of 12:01, 18 September 2023

Problem

Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:

\[\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}\]

What is $a-d$?

$\text{(A) }4 \qquad \text{(B) }6 \qquad \text{(C) }8 \qquad \text{(D) }10 \qquad \text{(E) }12$

Solution

Using Simon's Favorite Factoring Trick, we can rewrite the three equations as follows:

\begin{align*} (a+1)(b+1) & = 525 \\ (b+1)(c+1) & = 147 \\ (c+1)(d+1) & = 105 \end{align*}

Let $(e,f,g,h)=(a+1,b+1,c+1,d+1)$. We get:

\begin{align*} ef & = 3\cdot 5\cdot 5\cdot 7 \\ fg & = 3\cdot 7\cdot 7 \\ gh & = 3\cdot 5\cdot 7 \end{align*}

Clearly $7^2$ divides $fg$. On the other hand, $7^2$ can not divide $f$, as it then would divide $ef$. Similarly, $7^2$ can not divide $g$. Hence $7$ divides both $f$ and $g$. This leaves us with only two cases: $(f,g)=(7,21)$ and $(f,g)=(21,7)$.

The first case solves to $(e,f,g,h)=(75,7,21,5)$, which gives us $(a,b,c,d)=(74,6,20,4)$, but then $abcd \not= 8!$. We do not need to multiply, it is enough to note e.g. that the left hand side is not divisible by $7$. (Also, a - d equals $70$ in this case, which is way too large to fit the answer choices.)

The second case solves to $(e,f,g,h)=(25,21,7,15)$, which gives us a valid quadruple $(a,b,c,d)=(24,20,6,14)$, and we have $a-d=24-14 =\boxed{10}$.

Video Solution

https://youtu.be/zlzievV5e-U

See Also

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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 12 Problems and Solutions

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