let-b-and-r-be-two-positive-prime-numbers-such-that-b-r-and-b-r-is-a-divisor-of-138-Consider-an-arithmetic-progression-in-which-the-first-term-is-b-the-ratio-is-r-and-the-fourth-term-is-71-What-i

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Question Number 74703 by Mr. K last updated on 29/Nov/19

$$let\mathit{b}and\mathit{r}betwopositiveprime$$$$numberssuchthatb\ne randb\times ris$$$$adivisorof138.Consideran$$$$arithmeticprogressioninwhich$$$$thefirsttermis\mathit{b},theratiois\mathit{r}$$$$andthefourthtermis71.Whatisthe$$$$valueof\mathit{b}+\mathit{r}?$$

Answered by mind is power last updated on 29/Nov/19

$$\mathrm{br}\mid 138$$$$138=2.69=2.3.23$$$$$$$$\mathrm{b}+3\mathrm{r}=71\Rightarrow \mathrm{b}\equiv -1\left(3\right)$$$$\Rightarrow \mathrm{b}\in \{2,23\}$$$$\mathrm{b}+\mathrm{r}=71-2\mathrm{r}\Rightarrow \mathrm{b}+\mathrm{r}=1\left(2\right)\Rightarrow \mathrm{b},\mathrm{r}\mathrm{are}\mathrm{in}\mathrm{different}\mathrm{parite}$$$$\mathrm{one}\mathrm{of}\mathrm{them}\mathrm{is}2$$$$\mathrm{b}=71-3\mathrm{r}⩽23\Rightarrow \mathrm{r}⩾\frac{71-23}{2}=\frac{48}{3}=16\Rightarrow \mathrm{r}=23,\mathrm{b}=2$$$$\mathrm{b}+\mathrm{r}=25$$$$$$$$$$$$$$$$$$$$$$

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