Question-157835

Question Number 157835 by HongKing last updated on 28/Oct/21

Answered by Rasheed.Sindhi last updated on 29/Oct/21

abcd^(−) : a<b<c<d ∧ a+b+c+d≡1[3]∧ a+b+c+d≡5[11]_(−) •1234≤abcd^(−) ≤6789 (1+2+3+4)≤(a+b+c+d)≤6+7+8+9) •10≤(a+b+c+d)≤30 ; •Also 1≤a≤6 [ ∵ The numbers are 4-digit] 2≤b≤7 3≤c≤8 4≤d≤9 ▶a+b+c+d≡5[11]:a+b+c+d=16,27 Only 16 satisfies a+b+c+d≡1[3] • ∴ a+b+c+d=16 determinant ((((i)a<b<c<d (ii)a+b+c+d=16))) 12:123_ ,1249 , 1258,1267,127_ 13:1348,1357,136_, 14:1456,146_, 15:156_, 16:167_ 17:178_ 23:2347,2356,236_ 24:245_, 25:256_ 26:267_ 27:278_ 34:345_ 35:356_ 36:367_ 37:378_ 45:456_ 46:467_ 47:478_ 56:567_ 57:578_ 67:678_ ′_′ means there′s no proper digit Required numbers: 1249 , 1258,1267,1348,1357,1456, 2347,2356

$$\overline {{abcd}}\::\:\:{a}<{b}<{c}<{d}\: \\ $$$$\underset{−} {\wedge\:{a}+{b}+{c}+{d}\equiv\mathrm{1}\left[\mathrm{3}\right]\wedge\:{a}+{b}+{c}+{d}\equiv\mathrm{5}\left[\mathrm{11}\right]} \\ $$$$\bullet\mathrm{1234}\leqslant\overline {{abcd}}\leqslant\mathrm{6789} \\ $$$$\left.\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}\right)\leqslant\left({a}+{b}+{c}+{d}\right)\leqslant\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}\right) \\ $$$$\bullet\mathrm{10}\leqslant\left({a}+{b}+{c}+{d}\right)\leqslant\mathrm{30}\:;\: \\ $$$$\:\bullet{Also}\:\mathrm{1}\leqslant{a}\leqslant\mathrm{6}\:\left[\:\because\:\mathcal{T}{he}\:{numbers}\:{are}\:\mathrm{4}-{digit}\right] \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{2}\leqslant{b}\leqslant\mathrm{7} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{3}\leqslant{c}\leqslant\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{4}\leqslant{d}\leqslant\mathrm{9} \\ $$$$\blacktriangleright{a}+{b}+{c}+{d}\equiv\mathrm{5}\left[\mathrm{11}\right]:{a}+{b}+{c}+{d}=\mathrm{16},\mathrm{27} \\ $$$$\:\:\:\:{Only}\:\mathrm{16}\:{satisfies}\:{a}+{b}+{c}+{d}\equiv\mathrm{1}\left[\mathrm{3}\right] \\ $$$$\bullet\:\:\therefore\:{a}+{b}+{c}+{d}=\mathrm{16} \\ $$$$\begin{array}{|c|}{\left({i}\right){a}<{b}<{c}<{d}\:\:\left({ii}\right){a}+{b}+{c}+{d}=\mathrm{16}}\\\hline\end{array} \\ $$$$\mathrm{12}:\mathrm{123\_}\:\:,\mathrm{1249}\:,\:\mathrm{1258},\mathrm{1267},\mathrm{127\_} \\ $$$$\mathrm{13}:\mathrm{1348},\mathrm{1357},\mathrm{136\_}, \\ $$$$\mathrm{14}:\mathrm{1456},\mathrm{146\_}, \\ $$$$\mathrm{15}:\mathrm{156\_}, \\ $$$$\mathrm{16}:\mathrm{167\_} \\ $$$$\mathrm{17}:\mathrm{178\_} \\ $$$$\mathrm{23}:\mathrm{2347},\mathrm{2356},\mathrm{236\_} \\ $$$$\mathrm{24}:\mathrm{245\_}, \\ $$$$\mathrm{25}:\mathrm{256\_} \\ $$$$\mathrm{26}:\mathrm{267\_} \\ $$$$\mathrm{27}:\mathrm{278\_} \\ $$$$\mathrm{34}:\mathrm{345\_} \\ $$$$\mathrm{35}:\mathrm{356\_} \\ $$$$\mathrm{36}:\mathrm{367\_} \\ $$$$\mathrm{37}:\mathrm{378\_} \\ $$$$\mathrm{45}:\mathrm{456\_} \\ $$$$\mathrm{46}:\mathrm{467\_} \\ $$$$\mathrm{47}:\mathrm{478\_} \\ $$$$\mathrm{56}:\mathrm{567\_} \\ $$$$\mathrm{57}:\mathrm{578\_} \\ $$$$\mathrm{67}:\mathrm{678\_} \\ $$$$'\_'\:{means}\:{there}'{s}\:{no}\:{proper}\:{digit} \\ $$$${Required}\:{numbers}: \\ $$$$\mathrm{1249}\:,\:\mathrm{1258},\mathrm{1267},\mathrm{1348},\mathrm{1357},\mathrm{1456}, \\ $$$$\mathrm{2347},\mathrm{2356} \\ $$

Commented by HongKing last updated on 28/Oct/21