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Question Number 191775 by Mingma last updated on 30/Apr/23

Answered by deleteduser1 last updated on 30/Apr/23

$$\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{ABCDEF}=10000\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{ABCD}+\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{EF}$$$$17[10000\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}+\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}]=12[100\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}+\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}]$$$$170000\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}-12\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}=1200\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}-17\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}$$$$\Rightarrow \frac{\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}}{\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}}=\frac{1183}{169988}=\frac{13}{1868}$$$$Notethat\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}}{AB}and\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{CDEF}canstillbeincreased$$$$byafactorof5while\stackrel{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{ABCDEF}remainsa$$$$6-digitnumber$$$$\Rightarrow Thelargest6-digitnumber=659340$$$$Thesmallest6-digitnumber=131868$$

Commented by Mingma last updated on 30/Apr/23

Excellent!

Answered by Frix last updated on 30/Apr/23

$$\mathrm{This}\mathrm{leads}\mathrm{to}$$$$\frac{AB}{13}=\frac{CDEF}{1868}$$$$\mathrm{With}\mathrm{all}\mathrm{digits}\mathrm{different}\mathrm{we}\mathrm{have}\mathrm{only}2$$$$\mathrm{possibilities}:$$$$\frac{39}{13}=\frac{5604}{1868}=3$$$$\frac{65}{13}=\frac{9340}{1868}=5$$$$\Rightarrow$$$$\mathrm{Answer}\mathrm{is}659340$$

Commented by Mingma last updated on 30/Apr/23

Excellent!

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