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Question Number 137745 by bemath last updated on 06/Apr/21

$$ \\ $$ Each of the digits 2, 4, 6, and 8 can be used once and once only in writing a four-digit number. What is the sum of all such numbers that are divisible by 11?\\n

Answered by TheSupreme last updated on 06/Apr/21

ABCD is div for 11 if  A+C−B−D=0 or A+C−B−D=11  A+C=B+D  i: 8+2=4+6 is the only possible solution  A=(2,8),(4,6)  B=(4,6),(2,8)  C=(2,8),(4,6)  D=(4,6),(2,8)  n=4×2=8  Σ_(i=1) ^8 1000A_i +100B_i +10C_i +D_i =  =1000ΣA_i +100ΣB_i +10ΣC_i +ΣD_i   ΣA_i =40 [2+2+4+4+6+6+8+8]  =1111×40=44440

$${ABCD}\:{is}\:{div}\:{for}\:\mathrm{11}\:{if} \\ $$ $${A}+{C}−{B}−{D}=\mathrm{0}\:{or}\:{A}+{C}−{B}−{D}=\mathrm{11} \\ $$ $${A}+{C}={B}+{D} \\ $$ $${i}:\:\mathrm{8}+\mathrm{2}=\mathrm{4}+\mathrm{6}\:{is}\:{the}\:{only}\:{possible}\:{solution} \\ $$ $${A}=\left(\mathrm{2},\mathrm{8}\right),\left(\mathrm{4},\mathrm{6}\right) \\ $$ $${B}=\left(\mathrm{4},\mathrm{6}\right),\left(\mathrm{2},\mathrm{8}\right) \\ $$ $${C}=\left(\mathrm{2},\mathrm{8}\right),\left(\mathrm{4},\mathrm{6}\right) \\ $$ $${D}=\left(\mathrm{4},\mathrm{6}\right),\left(\mathrm{2},\mathrm{8}\right) \\ $$ $${n}=\mathrm{4}×\mathrm{2}=\mathrm{8} \\ $$ $$\underset{{i}=\mathrm{1}} {\overset{\mathrm{8}} {\sum}}\mathrm{1000}{A}_{{i}} +\mathrm{100}{B}_{{i}} +\mathrm{10}{C}_{{i}} +{D}_{{i}} = \\ $$ $$=\mathrm{1000}\Sigma{A}_{{i}} +\mathrm{100}\Sigma{B}_{{i}} +\mathrm{10}\Sigma{C}_{{i}} +\Sigma{D}_{{i}} \\ $$ $$\Sigma{A}_{{i}} =\mathrm{40}\:\left[\mathrm{2}+\mathrm{2}+\mathrm{4}+\mathrm{4}+\mathrm{6}+\mathrm{6}+\mathrm{8}+\mathrm{8}\right] \\ $$ $$=\mathrm{1111}×\mathrm{40}=\mathrm{44440} \\ $$

Commented byotchereabdullai@gmail.com last updated on 06/Apr/21

thank you sir!

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}! \\ $$

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