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Question-177040




Question Number 177040 by cortano1 last updated on 30/Sep/22
Answered by mr W last updated on 30/Sep/22
total number of 9−digit numbers is:  9!=362880    number of those which are divisible  by 4 is:  last digit is 2:   xxxxxxx12/32/52/72/92 ⇒ 5×7!  last digit is 4:   xxxxxxx24/64/84 ⇒3×7!  last digit is 6:   xxxxxxx16/36/56/76/96 ⇒5×7!  last digit is 8:   xxxxxxx28/48/68 ⇒3×7!  Σ: 16×7!=80640
$${total}\:{number}\:{of}\:\mathrm{9}−{digit}\:{numbers}\:{is}: \\ $$$$\mathrm{9}!=\mathrm{362880} \\ $$$$ \\ $$$${number}\:{of}\:{those}\:{which}\:{are}\:{divisible} \\ $$$${by}\:\mathrm{4}\:{is}: \\ $$$${last}\:{digit}\:{is}\:\mathrm{2}:\: \\ $$$$\mathrm{xxxxxxx12}/\mathrm{32}/\mathrm{52}/\mathrm{72}/\mathrm{92}\:\Rightarrow\:\mathrm{5}×\mathrm{7}! \\ $$$${last}\:{digit}\:{is}\:\mathrm{4}:\: \\ $$$$\mathrm{xxxxxxx24}/\mathrm{64}/\mathrm{84}\:\Rightarrow\mathrm{3}×\mathrm{7}! \\ $$$${last}\:{digit}\:{is}\:\mathrm{6}:\: \\ $$$$\mathrm{xxxxxxx16}/\mathrm{36}/\mathrm{56}/\mathrm{76}/\mathrm{96}\:\Rightarrow\mathrm{5}×\mathrm{7}! \\ $$$${last}\:{digit}\:{is}\:\mathrm{8}:\: \\ $$$$\mathrm{xxxxxxx28}/\mathrm{48}/\mathrm{68}\:\Rightarrow\mathrm{3}×\mathrm{7}! \\ $$$$\Sigma:\:\mathrm{16}×\mathrm{7}!=\mathrm{80640} \\ $$
Commented by cortano1 last updated on 30/Sep/22
why sir last digit is 2= 4×7!?
$$\mathrm{why}\:\mathrm{sir}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{is}\:\mathrm{2}=\:\mathrm{4}×\mathrm{7}!? \\ $$
Commented by mr W last updated on 30/Sep/22
i had some typos. now fixed.
$${i}\:{had}\:{some}\:{typos}.\:{now}\:{fixed}. \\ $$
Commented by Tawa11 last updated on 03/Oct/22
Great sir
$$\mathrm{Great}\:\mathrm{sir} \\ $$

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