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a and b are positive integers such that ab + 5a + b = 2000.  Find b.

May 7, 2020

#1
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a =4    and     b=396

May 7, 2020
#2
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a and b are positive integers such that $$ab + 5a + b = 2000$$.
Find b.

$$\begin{array}{|rcll|} \hline ab + 5a + b &=& 2000 \\ (a+1)(b+5)-5 &=& 2000 \\ \mathbf{(a+1)*(b+5)} &=& \mathbf{2005} \\ && \text{find all integer products }\ (a+1)*(b+5) = 2005 \\ \hline \end{array}$$

All integer products:

$$\begin{array}{|ccl|r|r|r|r|l|} \hline && (a+1)*(b+5) & (a+1) & (b+5) & a & b & \text{a and b are positive integers} \\ \hline 2005 &=& 1\times 2005 & 1 & 2005 & 0 & 2000 \\ 2005 &=& 5\times 401 & 5 & 401 & \mathbf{4} & \mathbf{396} & \checkmark \\ 2005 &=& 401\times 5 & 401 & 5 & 400 & 0 \\ 2005 &=& 2005\times 1 & 2005 & 1 & 2004 & -4 \\ \hline \end{array}$$

$$\text{b+5 = 401, so b=401-5 hence \mathbf{b = 396} }$$

May 8, 2020