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Question Number 110715 by Aina Samuel Temidayo last updated on 30/Aug/20
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.
$$\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:\mathrm{are}\:\mathrm{unit}\:\mathrm{digits}\:\mathrm{whose} \\ $$$$\mathrm{pairwise}\:\mathrm{sums}\:\mathrm{form}\:\mathrm{an}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}\:\mathrm{is} \\ $$$$\mathrm{even},\:\mathrm{find}\:\mathrm{the}\:\mathrm{common}\:\mathrm{positive} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arithmetic} \\ $$$$\mathrm{progression}. \\ $$
Commented by Rasheed.Sindhi last updated on 30/Aug/20
“pairwise sums of a,b,c,d” All possible pairs?Order?
$$“\mathrm{pairwise}\:\mathrm{sums}\:\mathrm{of}\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}'' \\ $$$${All}\:{possible}\:{pairs}?{Order}? \\ $$
Commented by Aina Samuel Temidayo last updated on 30/Aug/20
I didn′t change the question. That′s how it is.
$$\mathrm{I}\:\mathrm{didn}'\mathrm{t}\:\mathrm{change}\:\mathrm{the}\:\mathrm{question}.\:\mathrm{That}'\mathrm{s} \\ $$$$\mathrm{how}\:\mathrm{it}\:\mathrm{is}. \\ $$
Answered by Rasheed.Sindhi last updated on 30/Aug/20
a+b,b+c,c+d are in AP The common difference: (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 a,b,c_(−) ,d^(⌢) a+d=b+c is only codition for your question,if I undestand the question. a+d=b+c _( a+d∈E⇒b+c∈E⇒a+b+c+d∈E)^(•a+d∈O⇒b+c∈O⇒a+b+c+d∈E) a+b+c+d∈E −−−−−−− ^• d=b+c−a a,b,c,b+c−a b−a=b+c−a−c b−a=b−a ∴ a+b , b+c , c+d is an AP Generally common difference is c−a or d−b.For making it positive we will take c>a and d>b
$${a}+{b},{b}+{c},{c}+{d}\:{are}\:{in}\:{AP} \\ $$$${The}\:{common}\:{difference}: \\ $$$$\left({b}+{c}\right)−\left({a}+{b}\right)=\left({c}+{d}\right)−\left({b}+{c}\right) \\ $$$${c}−{a}={d}−{b} \\ $$$${b}+{c}={a}+{d} \\ $$$$−−−−− \\ $$$$\:\mathrm{1}+\mathrm{3}=\mathrm{0}+\mathrm{4}\:\:\:\:\:\: \\ $$$$\mathrm{0},\mathrm{1},\mathrm{3},\mathrm{4}\:\:\:\:\:\:\: \\ $$$$\mathrm{0}+\mathrm{1},\mathrm{1}+\mathrm{3},\mathrm{3}+\mathrm{4} \\ $$$$\mathrm{1},\mathrm{4},\mathrm{7} \\ $$$$−−−−− \\ $$$$\mathrm{0},\mathrm{2},\mathrm{3},\mathrm{5} \\ $$$$\mathrm{0}+\mathrm{2},\mathrm{2}+\mathrm{3},\mathrm{3}+\mathrm{5} \\ $$$$\mathrm{2},\mathrm{5},\mathrm{8} \\ $$$$−−−−−− \\ $$$$\mathrm{0},\mathrm{2},\mathrm{4},\mathrm{6} \\ $$$$\mathrm{0}+\mathrm{2},\mathrm{2}+\mathrm{4},\mathrm{4}+\mathrm{6} \\ $$$$\mathrm{2},\mathrm{6},\mathrm{10} \\ $$$$−−−−− \\ $$$$\mathrm{1},\mathrm{3},\mathrm{6},\mathrm{8} \\ $$$$\mathrm{1}+\mathrm{3},\mathrm{3}+\mathrm{6},\mathrm{6}+\mathrm{8} \\ $$$$\mathrm{4},\mathrm{9},\mathrm{14} \\ $$$$−−−−−− \\ $$$$\mathrm{1},\mathrm{3},\mathrm{7},\mathrm{9} \\ $$$$\mathrm{4},\mathrm{10},\mathrm{16} \\ $$$$\overset{\frown} {{a},{b},{c},{d}}\:\:\:{a}+{d}={b}+{c} \\ $$$${is}\:{only}\:{codition}\:{for}\:{your} \\ $$$${question},{if}\:{I}\:{undestand}\:{the} \\ $$$${question}. \\ $$$${a}+{d}={b}+{c} \\ $$$$\:_{\:\:\:\:{a}+{d}\in\mathbb{E}\Rightarrow{b}+{c}\in\mathbb{E}\Rightarrow{a}+{b}+{c}+{d}\in\mathbb{E}} ^{\bullet{a}+{d}\in\mathbb{O}\Rightarrow{b}+{c}\in\mathbb{O}\Rightarrow{a}+{b}+{c}+{d}\in\mathbb{E}} \\ $$$$\:\:\:\:{a}+{b}+{c}+{d}\in\mathbb{E} \\ $$$$−−−−−−− \\ $$$$\:^{\bullet} {d}={b}+{c}−{a} \\ $$$$\:\:{a},{b},{c},{b}+{c}−{a} \\ $$$$\:\:\:{b}−{a}={b}+{c}−{a}−{c} \\ $$$$\:\:\:{b}−{a}={b}−{a} \\ $$$$\:\:\:\therefore\:{a}+{b}\:,\:{b}+{c}\:,\:{c}+{d}\:{is}\:{an}\:{AP} \\ $$$${Generally}\:{common}\:{difference} \\ $$$$\:{is}\:{c}−{a}\:\:\:{or}\:\:{d}−{b}.{For}\:{making} \\ $$$${it}\:{positive}\:{we}\:{will}\:{take}\:{c}>{a}\:{and} \\ $$$${d}>{b} \\ $$
Commented by Aina Samuel Temidayo last updated on 30/Aug/20
So what is the common difference?
$$\mathrm{So}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference}? \\ $$
Commented by Rasheed.Sindhi last updated on 30/Aug/20
Common difference is not same for all solutions.Please read last lines in red color of my solution.
$${Common}\:{difference}\:{is}\:{not}\:{same} \\ $$$${for}\:{all}\:{solutions}.{Please}\:{read}\:{last} \\ $$$${lines}\:{in}\:{red}\:{color}\:{of}\:{my}\:{solution}. \\ $$
Commented by Aina Samuel Temidayo last updated on 30/Aug/20
With the options I have here, the common difference is only one.
$$\mathrm{With}\:\mathrm{the}\:\mathrm{options}\:\mathrm{I}\:\mathrm{have}\:\mathrm{here},\:\mathrm{the} \\ $$$$\mathrm{common}\:\mathrm{difference}\:\mathrm{is}\:\mathrm{only}\:\mathrm{one}. \\ $$

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