a-b-c-d-are-unit-digits-whose-pairwise-sums-form-an-arithmetic-progression-Given-that-a-b-c-d-is-even-find-the-common-positive-difference-of-the-arithmetic-progression-

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Question Number 110715 by Aina Samuel Temidayo last updated on 30/Aug/20

$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\mathrm{are}\mathrm{unit}\mathrm{digits}\mathrm{whose}$$$$\mathrm{pairwise}\mathrm{sums}\mathrm{form}\mathrm{an}\mathrm{arithmetic}$$$$\mathrm{progression}.\mathrm{Given}\mathrm{that}\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\mathrm{is}$$$$\mathrm{even},\mathrm{find}\mathrm{the}\mathrm{common}\mathrm{positive}$$$$\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{arithmetic}$$$$\mathrm{progression}.$$

Commented by Rasheed.Sindhi last updated on 30/Aug/20

$$“\mathrm{pairwise}\mathrm{sums}\mathrm{of}\mathrm{a},\mathrm{b},\mathrm{c},{\mathrm{d}}^{″}$$$$Allpossiblepairs?Order?$$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

$$\mathrm{I}{\mathrm{didn}}^{\prime }\mathrm{t}\mathrm{change}\mathrm{the}\mathrm{question}.{\mathrm{That}}^{\prime }\mathrm{s}$$$$\mathrm{how}\mathrm{it}\mathrm{is}.$$

Answered by Rasheed.Sindhi last updated on 30/Aug/20

$$a+b,b+c,c+dareinAP$$$$Thecommondifference:$$$$(b+c)-(a+b)=(c+d)-(b+c)$$$$c-a=d-b$$$$b+c=a+d$$$$-----$$$$1+3=0+4$$$$0,1,3,4$$$$0+1,1+3,3+4$$$$1,4,7$$$$-----$$$$0,2,3,5$$$$0+2,2+3,3+5$$$$2,5,8$$$$------$$$$0,2,4,6$$$$0+2,2+4,4+6$$$$2,6,10$$$$-----$$$$1,3,6,8$$$$1+3,3+6,6+8$$$$4,9,14$$$$------$$$$1,3,7,9$$$$4,10,16$$$$\stackrel{⌢}{a,b,c,d}a+d=b+c$$$$isonlycoditionforyour$$$$question,ifIundestandthe$$$$question.$$$$a+d=b+c$$$${}_{a+d\in \mathbb{E}\Rightarrow b+c\in \mathbb{E}\Rightarrow a+b+c+d\in \mathbb{E}}^{\bullet a+d\in \mathbb{O}\Rightarrow b+c\in \mathbb{O}\Rightarrow a+b+c+d\in \mathbb{E}}$$$$a+b+c+d\in \mathbb{E}$$$$-------$$$${}^{\bullet }d=b+c-a$$$$a,b,c,b+c-a$$$$b-a=b+c-a-c$$$$b-a=b-a$$$$\therefore a+b,b+c,c+disanAP$$$$Generallycommondifference$$$$isc-aord-b.Formaking$$$$itpositivewewilltakec>aand$$$$d>b$$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

$$\mathrm{So}\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{common}\mathrm{difference}?$$

Commented by Rasheed.Sindhi last updated on 30/Aug/20

$$Commondifferenceisnotsame$$$$forallsolutions.Pleasereadlast$$$$linesinredcolorofmysolution.$$

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

$$\mathrm{With}\mathrm{the}\mathrm{options}\mathrm{I}\mathrm{have}\mathrm{here},\mathrm{the}$$$$\mathrm{common}\mathrm{difference}\mathrm{is}\mathrm{only}\mathrm{one}.$$

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