My Math Forum fill in the blank and find X,Y,Z

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July 18th, 2012, 07:38 AM   #1
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fill in the blank and find X,Y,Z

[attachment=0:295e7eqx]Quotient.jpg[/attachment:295e7eqx]
X=dividend
Y=divisor
Z=quotient
fill in the blank ,and find X,Y,Z
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 Quotient.jpg (12.1 KB, 186 views)

July 18th, 2012, 09:37 AM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

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Re: fill in the blank and find X,Y,Z

Hello, Albert.Teng!

This is a classic alphametic . . .

Quote:
 $\begin{array}{ccccccccccc} &&&&&&& a & b & 8 & c & d & [1] \\ &&&& -- & -- & -- & -- & -- & -- & -- & -- \\ x&x&x & | & x&x&x&x&x&x&x&x & [2] \\ &&&&& x&x&x &&&&& [3] \\ &&&& --&--&--&-- \\ &&&&&& x&x&x&x &&& [4]\\ &&&&&&& x&x&x &&& [5] \\ &&&&&& --&--&--&-- \\ &&&&&&&& x&x&x&x & [6]\\ &&&&&&&& x&x&x&x & [7]\\ &&&&&&&& --&--&--&-- \end{array}$

$\text{Let }D\,=\,\text{ the three-digit divisor.}$

$\text{We see that line [2] begins: }\,1\,0\,x\,x
\;\;\;\text{and line [3] is: }\,9\,x\,x
\text{The same is true for lines [4] and [5].}$

$\begin{array}{ccccccccccc}
&&&&& a & b & 8 & c & d & [1] \\
&& -- & -- & -- & -- & -- & -- & -- & -- \\
D & | & 1&0&x&x&x&x&x&x & [2] \\
&&& 9&x&x &&&&& [3] \\
&& --&--&--&-- \\
&&&& 1&0&x&x &&& [4]\\
&&&&& 9&x&x &&& [5] \\
&&&& --&--&--&-- \\
&&&&&& x&x&x&x & [6]\\
&&&&&& x&x&x&x & [7]\\
&&&&&& --&--&--&-- \end{array}$

$\text{In lines [4] and [6], we see that two digits were brought down.}
\;\;\;\text{Hence: }\,b\,=\,0,\;c\,=\,0$

$\begin{array}{ccccccccccc}
&&&&& a & 0 & 8 & 0 & d & [1] \\
&& -- & -- & -- & -- & -- & -- & -- & -- \\
D & | & 1&0&x&x&x&x&x&x & [2] \\
&&& 9&x&x &&&&& [3] \\
&& --&--&--&-- \\
&&&& 1&0&x&x &&& [4]\\
&&&&& 9&x&x &&& [5] \\
&&&& --&--&--&-- \\
&&&&&& x&x&x&x & [6]\\
&&&&&& x&x&x&x & [7]\\
&&&&&& --&--&--&-- \end{array}$

$\text{In line [5]: }\:8\,\times\,D\text{ is a three-digit number.}
\;\;\;\text{Hence: }\:8D\,<\,1000 \;\;\;\Rightarrow\;\;\;D\,<\,125$

$\text{In line [3], }\,a\,\times\,D \:=\:9\,x\,x
\;\;\;\text{Since }\,7\,\times\,D \:<\:875,\;a \,=\,8$

$\text{In line [7]: }d\,\times\,D\text{ is a four-digit number.}
\;\;\;\text{Hence: }\:d \,=\,9$

$\begin{array}{ccccccccccc}
&&&&& 8 & 0 & 8 & 0 & 9 & [1] \\
&& -- & -- & -- & -- & -- & -- & -- & -- \\
D & | & 1&0&x&x&x&x&x &x& [2] \\
&&& 9&x&x &&&&& [3] \\
&& --&--&--&-- \\
&&&& 1&0&x&x &&& [4]\\
&&&&& 9&x&x &&& [5] \\
&&&& --&--&--&-- \\
&&&&&& x&x&x&x & [6]\\
&&&&&& x&x&x&x & [7]\\
&&&&&& --&--&--&-- \end{array}$

$\text{W\!e see that: }\:80809\,\times\,D \:>\:10,000,000 \;\;\;D \:>\:123.748...$

$\text{Since }\,D\,\ge\,123\:\text{ and }\:D \,=<\,125,\,\text{ then: }\:D \:=\:124$

$\text{Solution:}$

$\begin{array}{ccccccccccc}
&&&&&&& 8 & 0 & 8 & 0 & 9 \\
&&&& -- & -- & -- & -- & -- & -- & -- & -- \\
1&2&4 & | & 1&0&0&2&0&3&1&6\\
&&&&& 9&9&2\\
&&&& --&--&--&-- \\
&&&&&& 1&0&0&3\\
&&&&&&& 9&9&2 \\
&&&&&& --&--&--&-- \\
&&&&&&&& 1&1&1&6 \\
&&&&&&&& 1&1&1&6\\
&&&&&&&& --&--&--&-- \end{array}$

 July 18th, 2012, 02:37 PM #3 Senior Member   Joined: Apr 2012 Posts: 799 Thanks: 1 Re: fill in the blank and find X,Y,Z soroban: what a marvelous solution ! you are so great !

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