Question-188147

Question Number 188147 by normans last updated on 26/Feb/23

Answered by Rasheed.Sindhi last updated on 26/Feb/23

5 ∣ ABC ⇒C=0 or 5 C=0⇒D=0,4,8 [∵ 4∣ABCD] C=5⇒D=2,6 [∵ 4∣ABCD] CD=00,04,08,52,56 BCD=000,900,504,108,252,756 [∵9∣BCD] B=0⇒A=3,6,9 [∵ 6∣AB ] B=9,5,1,7⇒A has no value [∵ 6∣AB ] B=2⇒A=1,4,7 [∵ 6∣AB ] A=1,3,4,6,7,9

$$\mathrm{5}\:\mid\:{ABC}\:\Rightarrow{C}=\mathrm{0}\:{or}\:\mathrm{5} \\ $$$${C}=\mathrm{0}\Rightarrow{D}=\mathrm{0},\mathrm{4},\mathrm{8}\:\:\left[\because\:\mathrm{4}\mid{ABCD}\right] \\ $$$${C}=\mathrm{5}\Rightarrow{D}=\mathrm{2},\mathrm{6}\:\:\:\:\:\:\left[\because\:\mathrm{4}\mid{ABCD}\right] \\ $$$${CD}=\mathrm{00},\mathrm{04},\mathrm{08},\mathrm{52},\mathrm{56} \\ $$$${BCD}=\mathrm{000},\mathrm{900},\mathrm{504},\mathrm{108},\mathrm{252},\mathrm{756}\:\:\left[\because\mathrm{9}\mid{BCD}\right] \\ $$$${B}=\mathrm{0}\Rightarrow{A}=\mathrm{3},\mathrm{6},\mathrm{9}\:\:\:\:\left[\because\:\mathrm{6}\mid{AB}\:\right] \\ $$$${B}=\mathrm{9},\mathrm{5},\mathrm{1},\mathrm{7}\Rightarrow{A}\:{has}\:{no}\:{value}\:\left[\because\:\mathrm{6}\mid{AB}\:\right] \\ $$$${B}=\mathrm{2}\Rightarrow{A}=\mathrm{1},\mathrm{4},\mathrm{7}\:\:\:\:\:\:\left[\because\:\mathrm{6}\mid{AB}\:\right] \\ $$$$ \\ $$$${A}=\mathrm{1},\mathrm{3},\mathrm{4},\mathrm{6},\mathrm{7},\mathrm{9} \\ $$

Commented by JDamian last updated on 26/Feb/23

B and D are even ⇒ BCD ∉{108,504,900} C=5 because B+C+D must be either 9 or 18

$${B}\:\mathrm{and}\:{D}\:\mathrm{are}\:\mathrm{even}\:\:\Rightarrow\:\:{BCD}\:\notin\left\{\mathrm{108},\mathrm{504},\mathrm{900}\right\} \\ $$$${C}=\mathrm{5}\:\:\mathrm{because}\:\:\mathrm{B}+{C}+{D}\:\:\mathrm{must}\:\mathrm{be}\:\mathrm{either}\:\mathrm{9}\:\mathrm{or}\:\mathrm{18} \\ $$

Commented by Rasheed.Sindhi last updated on 26/Feb/23

$${My}\:{mistake}\:{sir},{thank}\:{you}! \\ $$

Answered by mr W last updated on 26/Feb/23

i think there are three such numbers: 1252, 4252, 7252

$${i}\:{think}\:{there}\:{are}\:{three}\:{such}\:{numbers}: \\ $$$$\mathrm{1252},\:\mathrm{4252},\:\mathrm{7252} \\ $$

Commented by mr W last updated on 26/Feb/23

ABCD B=even, ≤8 C=0 or 5 B+C+D=9 or 18 if C=0: B+D=9 ⇒D=odd ⇒rejected B+D=18 ⇒D≥10 ⇒impossible if C=5: B+D=4 ⇒(B, D)=(0,4) or (4,0) or (2,2) B+D=13 ⇒D=odd ⇒rejected A054, A450, A252 are remaining. A054: A=6 or 9 ⇒not divisible by 4 ⇒rejected A450: A=2 or 5 or 8 ⇒not divisible by 4 ⇒rejected A252: A=1 or 4 or 7 ⇒divisible by 4 ⇒ok so the number is 1252, 4252 or 7252.

$${ABCD} \\ $$$${B}={even},\:\leqslant\mathrm{8} \\ $$$${C}=\mathrm{0}\:{or}\:\mathrm{5} \\ $$$${B}+{C}+{D}=\mathrm{9}\:{or}\:\mathrm{18} \\ $$$${if}\:{C}=\mathrm{0}:\:{B}+{D}=\mathrm{9}\:\Rightarrow{D}={odd}\:\Rightarrow{rejected} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{B}+{D}=\mathrm{18}\:\Rightarrow{D}\geqslant\mathrm{10}\:\Rightarrow{impossible} \\ $$$${if}\:{C}=\mathrm{5}:\:{B}+{D}=\mathrm{4}\:\Rightarrow\left({B},\:{D}\right)=\left(\mathrm{0},\mathrm{4}\right)\:{or}\:\left(\mathrm{4},\mathrm{0}\right)\:{or}\:\left(\mathrm{2},\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{B}+{D}=\mathrm{13}\:\Rightarrow{D}={odd}\:\Rightarrow{rejected} \\ $$$${A}\mathrm{054},\:{A}\mathrm{450},\:{A}\mathrm{252}\:{are}\:{remaining}. \\ $$$${A}\mathrm{054}:\:{A}=\mathrm{6}\:{or}\:\mathrm{9}\:\Rightarrow{not}\:{divisible}\:{by}\:\mathrm{4}\:\Rightarrow{rejected} \\ $$$${A}\mathrm{450}:\:{A}=\mathrm{2}\:{or}\:\mathrm{5}\:{or}\:\mathrm{8}\:\Rightarrow{not}\:{divisible}\:{by}\:\mathrm{4}\:\Rightarrow{rejected} \\ $$$${A}\mathrm{252}:\:{A}=\mathrm{1}\:{or}\:\mathrm{4}\:{or}\:\mathrm{7}\:\Rightarrow{divisible}\:{by}\:\mathrm{4}\:\Rightarrow{ok} \\ $$$${so}\:{the}\:{number}\:{is}\:\mathrm{1252},\:\mathrm{4252}\:{or}\:\mathrm{7252}. \\ $$

Answered by manxsol last updated on 26/Feb/23

determinant ((A,B,C,D,),(,,0,,5),(,,5,,5),(,5,0,4,(4,9)),(,1,0,8,(4,9)),(,9,0,0,(4,9)),(,2,5,2,(4,9)),(,7,5,6,(4,9)),(,2,5,2,(6=2∧3)),((1,4,7),2,5,2,3))

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}{{A}}&\hline{{B}}&\hline{{C}}&\hline{{D}}&\hline{}\\{}&\hline{}&\hline{\mathrm{0}}&\hline{}&\hline{\mathrm{5}}\\{}&\hline{}&\hline{\mathrm{5}}&\hline{}&\hline{\mathrm{5}}\\{}&\hline{\mathrm{5}}&\hline{\mathrm{0}}&\hline{\mathrm{4}}&\hline{\mathrm{4},\mathrm{9}}\\{}&\hline{\mathrm{1}}&\hline{\mathrm{0}}&\hline{\mathrm{8}}&\hline{\mathrm{4},\mathrm{9}}\\{}&\hline{\mathrm{9}}&\hline{\mathrm{0}}&\hline{\mathrm{0}}&\hline{\mathrm{4},\mathrm{9}}\\{}&\hline{\mathrm{2}}&\hline{\mathrm{5}}&\hline{\mathrm{2}}&\hline{\mathrm{4},\mathrm{9}}\\{}&\hline{\mathrm{7}}&\hline{\mathrm{5}}&\hline{\mathrm{6}}&\hline{\mathrm{4},\mathrm{9}}\\{}&\hline{\mathrm{2}}&\hline{\mathrm{5}}&\hline{\mathrm{2}}&\hline{\mathrm{6}=\mathrm{2}\wedge\mathrm{3}}\\{\mathrm{1},\mathrm{4},\mathrm{7}}&\hline{\mathrm{2}}&\hline{\mathrm{5}}&\hline{\mathrm{2}}&\hline{\mathrm{3}}\\\hline\end{array} \\ $$$$ \\ $$

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