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There are four unequal, positive integers a, b, c, and N such that N = 5a + 3b + 5c . It is also true that N = 4a + 5b + 4c and N is between 131 and 150. What is the value of a + b + c ?

Jun 4, 2020

#1
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There are four unequal, positive integersa, b, candNsuch thatN= 5a+ 3b+ 5c.
It is also true that N= 4a+ 5b+ 4c and N is between 131 and 150.
What is the value of a+b+c?

$$\begin{array}{|rcll|} \hline N &=& 5a+ 3b+ 5c \\ -N &=& 4a+ 5b+ 4c \\ \hline 0 &=& a-2b+c \qquad \text{or} \qquad \mathbf{a+c= 2b} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline N &=& 5a+ 3b+ 5c \\ N &=& 5(a+c) + 3b \quad | \quad a+c= 2b \\ N &=& 5*2b + 3b \\ \mathbf{N} &=& \mathbf{13b} \qquad \text{, where b is an integer} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \text{Multiples of 13 between 131 and 150:} \\ \hline 13\times10 &=& 130 \\ 13\times11 &=& 143\ \leftarrow \ \text{ the only one} \\ 13\times12 &=& 156 \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{N} &=& \mathbf{13b} \quad | \quad \mathbf{N=143} \\ 143 &=& 13b \quad | \quad : 13 \\ 11 &=& b \\ \mathbf{b} &=& \mathbf{11} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{a+c} &=& \mathbf{2b} \quad | \quad \mathbf{b=11} \\ a+c &=& 2*11 \\ \mathbf{a+c} &=& \mathbf{22} \\ \hline a+b+c &=& (a+c) +b \\ a+b+c &=& 22 +11 \\ \mathbf{a+b+c} &=& \mathbf{33} \\ \hline \end{array}$$

Jun 4, 2020