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# Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and $a + n = b - n = \frac{c}{n},$ compute the value of $a$.

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1408
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Let $a$, $b$, $c$, and $n$ be positive integers. If $a + b + c = 19 \cdot 97$ and $a + n = b - n = \frac{c}{n},$ compute the value of $a$.

$$Let a, b, c, and n be positive integers. If a + b + c = 19 \cdot 97 and $a + n = b - n = \frac{c}{n},$ compute the value of a.$$

Mellie  Jul 2, 2015

#2
+20025
+15

$$\small{\text{ \begin{array}{lcl} Let  a, b, c,  and  n  be positive integers. If  a + b + c = 19 \cdot 97  and  \\ \left[a + n = b - n = \dfrac{c}{n}\right],  compute the value of  a . \end{array} }}$$

$$\small{\text{ \begin{array}{lrrrcl} & a+b+c=19\cdot 97 \\ \\ \hline \\ (1)& a+n=k \\ (2) & b-n=k &\qquad \qquad (1)+(2): & a+b&=& 2\cdot k\\ (3) & \dfrac{c}{n}=k &\qquad \qquad so & c &=& k\cdot n\\\\ & & & a+b+c &=& 2\cdot k + k\cdot n=19\cdot 97 \\ \hline \\ \end{array} }}\\\\ \small{\text{ \begin{array}{rrclrcl} & 2\cdot k + k\cdot n &=& 19\cdot 97\\\\ & k\cdot(2+n)&=& 19\cdot 97\\\\ I. & {k}\cdot{(2+n)}&=& {19}\cdot {97}\\\\ & \underline{k=19} && \underline{2+n = 97 }& \qquad \Rightarrow \qquad n&=& 95\\ & && & a+n&=& k\\ & && & a+95&=& 19\\ & && & a&=& -76 ~ negative! \\ \\ II. & {k}\cdot{(2+n)}&=& {97}\cdot {19}\\\\ & \underline{k=97} && \underline{2+n = 19 }& \qquad \Rightarrow \qquad n&=& 17\\ & && & a+n&=& k\\ & && & a+17&=& 97\\ & && & a&=& 80 ~ okay! \\ \end{array} }}$$

heureka  Jul 3, 2015
#1
+27044
+15

.

Alan  Jul 2, 2015
#2
+20025
+15

$$\small{\text{ \begin{array}{lcl} Let  a, b, c,  and  n  be positive integers. If  a + b + c = 19 \cdot 97  and  \\ \left[a + n = b - n = \dfrac{c}{n}\right],  compute the value of  a . \end{array} }}$$

$$\small{\text{ \begin{array}{lrrrcl} & a+b+c=19\cdot 97 \\ \\ \hline \\ (1)& a+n=k \\ (2) & b-n=k &\qquad \qquad (1)+(2): & a+b&=& 2\cdot k\\ (3) & \dfrac{c}{n}=k &\qquad \qquad so & c &=& k\cdot n\\\\ & & & a+b+c &=& 2\cdot k + k\cdot n=19\cdot 97 \\ \hline \\ \end{array} }}\\\\ \small{\text{ \begin{array}{rrclrcl} & 2\cdot k + k\cdot n &=& 19\cdot 97\\\\ & k\cdot(2+n)&=& 19\cdot 97\\\\ I. & {k}\cdot{(2+n)}&=& {19}\cdot {97}\\\\ & \underline{k=19} && \underline{2+n = 97 }& \qquad \Rightarrow \qquad n&=& 95\\ & && & a+n&=& k\\ & && & a+95&=& 19\\ & && & a&=& -76 ~ negative! \\ \\ II. & {k}\cdot{(2+n)}&=& {97}\cdot {19}\\\\ & \underline{k=97} && \underline{2+n = 19 }& \qquad \Rightarrow \qquad n&=& 17\\ & && & a+n&=& k\\ & && & a+17&=& 97\\ & && & a&=& 80 ~ okay! \\ \end{array} }}$$

heureka  Jul 3, 2015
#3
+93664
+5

2 great answers - thanks Alan and Heureka

Melody  Jul 3, 2015