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-t

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

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

Commented by malwaan last updated on 19/Apr/20

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

Answered by malwaan last updated on 20/Apr/20

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

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