Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 50732 by hassentimol last updated on 19/Dec/18

Answered by Rasheed.Sindhi last updated on 19/Dec/18

(10a+b)×c=10d+e (10d+e)+(10f+g)=10h+i • c can′t be 1 because in that case a=d & b=e •12≤10a+b≤49 (b/c c is at least 2) Continue

$$\left(\mathrm{10}{a}+{b}\right)×{c}=\mathrm{10}{d}+{e} \\ $$$$\left(\mathrm{10}{d}+{e}\right)+\left(\mathrm{10}{f}+{g}\right)=\mathrm{10}{h}+{i} \\ $$$$\bullet\:{c}\:{can}'{t}\:{be}\:\mathrm{1}\:{because}\:{in}\:{that}\:{case} \\ $$$${a}={d}\:\&\:{b}={e} \\ $$$$\bullet\mathrm{12}\leqslant\mathrm{10}{a}+{b}\leqslant\mathrm{49}\:\left({b}/{c}\:\:{c}\:{is}\:{at}\:{least}\:\mathrm{2}\right) \\ $$$${Continue} \\ $$

Commented by hassentimol last updated on 19/Dec/18

Thank you Sir. God bless you.

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

Commented by hassentimol last updated on 19/Dec/18

But it is impossible to resolve... There is too much unknown and only 2 equations... ?

$$\mathrm{But}\:\mathrm{it}\:\mathrm{is}\:\mathrm{impossible}\:\mathrm{to}\:\mathrm{resolve}... \\ $$$$\mathrm{There}\:\mathrm{is}\:\mathrm{too}\:\mathrm{much}\:\mathrm{unknown}\:\mathrm{and}\:\mathrm{only} \\ $$$$\mathrm{2}\:\mathrm{equations}...\:? \\ $$

Answered by Rasheed.Sindhi last updated on 20/Dec/18

1 7 × 4 −−−− 6 8 + 2 5 −−−− 9 3

$$\:\:\:\:\:\mathrm{1}\:\:\mathrm{7} \\ $$$$×\:\:\:\:\mathrm{4} \\ $$$$−−−− \\ $$$$\:\:\:\:\:\mathrm{6}\:\:\mathrm{8} \\ $$$$+\:\mathrm{2}\:\:\mathrm{5} \\ $$$$−−−− \\ $$$$\:\:\:\:\mathrm{9}\:\:\:\mathrm{3} \\ $$

Commented by ajfour last updated on 20/Dec/18

Excellent Sir, any method?

$${Excellent}\:{Sir},\:{any}\:{method}? \\ $$

Commented by Rasheed.Sindhi last updated on 20/Dec/18

Thanks for encouraging Sir! The method is a mixture of intelligent guessing & trial.

$$\mathcal{T}{han}\Bbbk{s}\:{for}\:{encouraging}\:{Sir}! \\ $$$${The}\:{method}\:{is}\:{a}\:{mixture}\:{of}\:{intelligent} \\ $$$${guessing}\:\&\:{trial}. \\ $$

Answered by MJS last updated on 21/Dec/18

ab×c=de b,c ≠1 otherwise e=b∨e=c ⇒ c≥2 de<100∧c≥2 ⇒ a≤4 b,c ≠5 otherwise e=5∨e=0 (b,c)∉{(2,6),(4,6),(6,2),(6,4),(6,8),(8,6)} which combinations fit? c=2∧a=1 ⇒ b∈{7,8,9} c=2∧a=3 ⇒ b∈{4,8,9} c=2∧a=4 ⇒ b∈{3,8} c=3∧a=1 ⇒ b∈{6,8,9} c=3∧a=2 ⇒ b∈{6,7,9} c=4∧a=1 ⇒ b∈{3,7,8,9} c=6∧a=1 ⇒ b=3 c=7∧a=1 ⇒ b∈{2,4} c=8∧a=1 ⇒ b=2 so we have: (1) listed c×de=fg (2) wrote down left digits in {} (3) “X” sorted out because no possibility to add a 2 digit number and stay <100 (4) testing the matches for de+fg=hi 2×17=34 {5,6,8,9} 34+f5=h9 X 2×18=36 {4,5,7,9} 36+f9=h5 X 2×19=38 {4,5,6,7} 38+f6=h4 X 38+f7=h5 X 2×34=68 {1,5,7,9} 68+f1=g9 X 68+f7=h5 X 68+f9=h7 X 2×38=76 {1,4,5,9} 76+f5=h1 X 76+f9=h5 X 2×39=78 {1,4,5,6} 78+f6=h4 X 2×43=86 {1,5,7,9} X 2×48=96 {1,3,5,7} X 3×16=48 {2,5,7,9} 48+f7=h5 X 48+f9=h7 X 3×18=54 {2,6,7,9} 54+f2=h6 X 3×19=57 {2,4,6,8} X 3×26=78 {1,4,5,9} 78+f1=h9 X 3×27=81 {4,5,6,9} X 3×29=87 {1,4,5,6} X 4×13=52 {6,7,8,9} X 4×17=68 {2,3,5,9} 68+f5=h3 ⇒ 68+25=93 4×18=72 {3,5,6,9} X 4×19=76 {2,3,5,8} 76+f2=h8 X 6×13=78 {2,4,5,9} X 7×12=84 {3,5,6,9} X 7×14=98 {2,3,5,6} X 8×12=96 {3,4,5,7} X

$${ab}×{c}={de} \\ $$$${b},{c}\:\neq\mathrm{1}\:\mathrm{otherwise}\:{e}={b}\vee{e}={c}\:\Rightarrow\:{c}\geqslant\mathrm{2} \\ $$$${de}<\mathrm{100}\wedge{c}\geqslant\mathrm{2}\:\Rightarrow\:{a}\leqslant\mathrm{4} \\ $$$${b},{c}\:\neq\mathrm{5}\:\mathrm{otherwise}\:{e}=\mathrm{5}\vee{e}=\mathrm{0} \\ $$$$\left({b},{c}\right)\notin\left\{\left(\mathrm{2},\mathrm{6}\right),\left(\mathrm{4},\mathrm{6}\right),\left(\mathrm{6},\mathrm{2}\right),\left(\mathrm{6},\mathrm{4}\right),\left(\mathrm{6},\mathrm{8}\right),\left(\mathrm{8},\mathrm{6}\right)\right\} \\ $$$$\mathrm{which}\:\mathrm{combinations}\:\mathrm{fit}? \\ $$$${c}=\mathrm{2}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}\in\left\{\mathrm{7},\mathrm{8},\mathrm{9}\right\} \\ $$$${c}=\mathrm{2}\wedge{a}=\mathrm{3}\:\Rightarrow\:{b}\in\left\{\mathrm{4},\mathrm{8},\mathrm{9}\right\} \\ $$$${c}=\mathrm{2}\wedge{a}=\mathrm{4}\:\Rightarrow\:{b}\in\left\{\mathrm{3},\mathrm{8}\right\} \\ $$$${c}=\mathrm{3}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}\in\left\{\mathrm{6},\mathrm{8},\mathrm{9}\right\} \\ $$$${c}=\mathrm{3}\wedge{a}=\mathrm{2}\:\Rightarrow\:{b}\in\left\{\mathrm{6},\mathrm{7},\mathrm{9}\right\} \\ $$$${c}=\mathrm{4}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}\in\left\{\mathrm{3},\mathrm{7},\mathrm{8},\mathrm{9}\right\} \\ $$$${c}=\mathrm{6}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}=\mathrm{3} \\ $$$${c}=\mathrm{7}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}\in\left\{\mathrm{2},\mathrm{4}\right\} \\ $$$${c}=\mathrm{8}\wedge{a}=\mathrm{1}\:\Rightarrow\:{b}=\mathrm{2} \\ $$$$\mathrm{so}\:\mathrm{we}\:\mathrm{have}: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{listed}\:{c}×{de}={fg} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{wrote}\:\mathrm{down}\:\mathrm{left}\:\mathrm{digits}\:\mathrm{in}\:\left\{\right\} \\ $$$$\left(\mathrm{3}\right)\:``{X}''\:\mathrm{sorted}\:\mathrm{out}\:\mathrm{because}\:\mathrm{no}\:\mathrm{possibility}\:\mathrm{to}\:\mathrm{add} \\ $$$$\mathrm{a}\:\mathrm{2}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{and}\:\mathrm{stay}\:<\mathrm{100} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{testing}\:\mathrm{the}\:\mathrm{matches}\:\mathrm{for}\:{de}+{fg}={hi} \\ $$$$\mathrm{2}×\mathrm{17}=\mathrm{34}\:\left\{\mathrm{5},\mathrm{6},\mathrm{8},\mathrm{9}\right\}\:\mathrm{34}+{f}\mathrm{5}={h}\mathrm{9}\:{X} \\ $$$$\mathrm{2}×\mathrm{18}=\mathrm{36}\:\left\{\mathrm{4},\mathrm{5},\mathrm{7},\mathrm{9}\right\}\:\mathrm{36}+{f}\mathrm{9}={h}\mathrm{5}\:{X} \\ $$$$\mathrm{2}×\mathrm{19}=\mathrm{38}\:\left\{\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7}\right\}\:\mathrm{38}+{f}\mathrm{6}={h}\mathrm{4}\:{X}\:\mathrm{38}+{f}\mathrm{7}={h}\mathrm{5}\:{X} \\ $$$$\mathrm{2}×\mathrm{34}=\mathrm{68}\:\left\{\mathrm{1},\mathrm{5},\mathrm{7},\mathrm{9}\right\}\:\mathrm{68}+{f}\mathrm{1}={g}\mathrm{9}\:{X}\:\mathrm{68}+{f}\mathrm{7}={h}\mathrm{5}\:{X}\:\mathrm{68}+{f}\mathrm{9}={h}\mathrm{7}\:{X} \\ $$$$\mathrm{2}×\mathrm{38}=\mathrm{76}\:\left\{\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{9}\right\}\:\mathrm{76}+{f}\mathrm{5}={h}\mathrm{1}\:{X}\:\mathrm{76}+{f}\mathrm{9}={h}\mathrm{5}\:{X} \\ $$$$\mathrm{2}×\mathrm{39}=\mathrm{78}\:\left\{\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{6}\right\}\:\mathrm{78}+{f}\mathrm{6}={h}\mathrm{4}\:{X} \\ $$$$\mathrm{2}×\mathrm{43}=\mathrm{86}\:\left\{\mathrm{1},\mathrm{5},\mathrm{7},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{2}×\mathrm{48}=\mathrm{96}\:\left\{\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{7}\right\}\:{X} \\ $$$$\mathrm{3}×\mathrm{16}=\mathrm{48}\:\left\{\mathrm{2},\mathrm{5},\mathrm{7},\mathrm{9}\right\}\:\mathrm{48}+{f}\mathrm{7}={h}\mathrm{5}\:{X}\:\mathrm{48}+{f}\mathrm{9}={h}\mathrm{7}\:{X} \\ $$$$\mathrm{3}×\mathrm{18}=\mathrm{54}\:\left\{\mathrm{2},\mathrm{6},\mathrm{7},\mathrm{9}\right\}\:\mathrm{54}+{f}\mathrm{2}={h}\mathrm{6}\:{X} \\ $$$$\mathrm{3}×\mathrm{19}=\mathrm{57}\:\left\{\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{8}\right\}\:{X} \\ $$$$\mathrm{3}×\mathrm{26}=\mathrm{78}\:\left\{\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{9}\right\}\:\mathrm{78}+{f}\mathrm{1}={h}\mathrm{9}\:{X} \\ $$$$\mathrm{3}×\mathrm{27}=\mathrm{81}\:\left\{\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{3}×\mathrm{29}=\mathrm{87}\:\left\{\mathrm{1},\mathrm{4},\mathrm{5},\mathrm{6}\right\}\:{X} \\ $$$$\mathrm{4}×\mathrm{13}=\mathrm{52}\:\left\{\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{4}×\mathrm{17}=\mathrm{68}\:\left\{\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{9}\right\}\:\mathrm{68}+{f}\mathrm{5}={h}\mathrm{3}\:\Rightarrow\:\mathrm{68}+\mathrm{25}=\mathrm{93} \\ $$$$\mathrm{4}×\mathrm{18}=\mathrm{72}\:\left\{\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{4}×\mathrm{19}=\mathrm{76}\:\left\{\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{8}\right\}\:\mathrm{76}+{f}\mathrm{2}={h}\mathrm{8}\:{X} \\ $$$$\mathrm{6}×\mathrm{13}=\mathrm{78}\:\left\{\mathrm{2},\mathrm{4},\mathrm{5},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{7}×\mathrm{12}=\mathrm{84}\:\left\{\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{9}\right\}\:{X} \\ $$$$\mathrm{7}×\mathrm{14}=\mathrm{98}\:\left\{\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{6}\right\}\:{X} \\ $$$$\mathrm{8}×\mathrm{12}=\mathrm{96}\:\left\{\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{7}\right\}\:{X} \\ $$

Commented by Rasheed.Sindhi last updated on 03/Feb/19

GooD strategy! Better than mine.

$$\mathcal{G}{oo}\mathcal{D}\:{strategy}!\:\mathcal{B}{etter}\:{than}\:{mine}. \\ $$

Commented by hassentimol last updated on 24/Dec/18

Thank you sir, amazing !

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir},\:\mathrm{amazing}\:! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com