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Question Number 89878 by malwaan last updated on 19/Apr/20

A four_digit whole number is interesting if the number formed by the leftmost two digits is twice as large as the number formed by the rightmost two digits. (for example 2010 is interesting) 1 find the largest whole number B such that all interesting numbers are divisible by B 2 find the smallest whole number D such that D is divisible by all interesting numbers.

$$\boldsymbol{{A}}\:\boldsymbol{{four\_digit}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{is}}\:\boldsymbol{{interesting}}\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{number}} \\ $$$$\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}}\:\boldsymbol{{leftmost}}\:\boldsymbol{{two}} \\ $$$$\boldsymbol{{digits}}\:\boldsymbol{{is}}\:\boldsymbol{{twice}}\:\boldsymbol{{as}}\:\boldsymbol{{large}}\:\boldsymbol{{as}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{number}}\:\boldsymbol{{formed}}\:\boldsymbol{{by}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{rightmost}}\:\boldsymbol{{two}}\:\boldsymbol{{digits}}. \\ $$$$\left(\boldsymbol{{for}}\:\boldsymbol{{example}}\:\mathrm{2010}\:\boldsymbol{{is}}\:\boldsymbol{{interesting}}\right) \\ $$$$\mathrm{1}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{largest}}\:\boldsymbol{{whole}}\:\boldsymbol{{number}} \\ $$$$\mathbb{B}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}\:\boldsymbol{{are}}\:\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\mathbb{B} \\ $$$$\mathrm{2}\:\:\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{smallest}}\:\boldsymbol{{whole}} \\ $$$$\boldsymbol{{number}}\:\mathbb{D}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:\mathbb{D}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{divisible}}\:\boldsymbol{{by}}\:\boldsymbol{{all}}\:\boldsymbol{{interesting}} \\ $$$$\boldsymbol{{numbers}}. \\ $$

Commented by malwaan last updated on 19/Apr/20

1 B = 201 why ?

$$\mathrm{1}\:\:\mathbb{B}\:=\:\mathrm{201}\:\:\:{why}\:? \\ $$

Answered by malwaan last updated on 20/Apr/20

0201=201×1 0402=201×2 . . 2010=201×10 2211=201×11 . . 9648=201×48 9849=201×49 ⇒201 is GCD ∴ B = 201

$$\mathrm{0201}=\mathrm{201}×\mathrm{1} \\ $$$$\mathrm{0402}=\mathrm{201}×\mathrm{2} \\ $$$$. \\ $$$$. \\ $$$$\mathrm{2010}=\mathrm{201}×\mathrm{10} \\ $$$$\mathrm{2211}=\mathrm{201}×\mathrm{11} \\ $$$$. \\ $$$$. \\ $$$$\mathrm{9648}=\mathrm{201}×\mathrm{48} \\ $$$$\mathrm{9849}=\mathrm{201}×\mathrm{49} \\ $$$$\Rightarrow\mathrm{201}\:\boldsymbol{{is}}\:\boldsymbol{{GCD}} \\ $$$$\therefore\:\mathbb{B}\:=\:\mathrm{201} \\ $$

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