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Question-193756




Question Number 193756 by Shlock last updated on 19/Jun/23
Answered by talminator2856792 last updated on 19/Jun/23
  7(7a + b) + c = 7(40) + 6        → c = 6 (mod 7)    → c = 6        7a + b = 40    since b is one digit,    difference between 40 and 40 − b is      a multiple of 7 not greater than 9;    only 7 × 1 applies.    → b = 40 − 7×⌊((40)/7)⌋          b = 5        → a = ⌊((40)/7)⌋           a = 5        cba − abc = 655 − 556    = 99
$$\:\:\mathrm{7}\left(\mathrm{7}{a}\:+\:{b}\right)\:+\:{c}\:=\:\mathrm{7}\left(\mathrm{40}\right)\:+\:\mathrm{6} \\ $$$$\:\: \\ $$$$\:\:\rightarrow\:{c}\:=\:\mathrm{6}\:\left(\mathrm{mod}\:\mathrm{7}\right) \\ $$$$\:\:\rightarrow\:{c}\:=\:\mathrm{6} \\ $$$$\:\: \\ $$$$\:\:\mathrm{7}{a}\:+\:{b}\:=\:\mathrm{40} \\ $$$$\:\:\mathrm{since}\:{b}\:\mathrm{is}\:\mathrm{one}\:\mathrm{digit}, \\ $$$$\:\:\mathrm{difference}\:\mathrm{between}\:\mathrm{40}\:\mathrm{and}\:\mathrm{40}\:−\:{b}\:\mathrm{is}\:\: \\ $$$$\:\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{7}\:\mathrm{not}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{9}; \\ $$$$\:\:\mathrm{only}\:\mathrm{7}\:×\:\mathrm{1}\:\mathrm{applies}. \\ $$$$\:\:\rightarrow\:{b}\:=\:\mathrm{40}\:−\:\mathrm{7}×\lfloor\frac{\mathrm{40}}{\mathrm{7}}\rfloor \\ $$$$\:\:\:\:\:\:\:\:{b}\:=\:\mathrm{5} \\ $$$$\:\: \\ $$$$\:\:\rightarrow\:{a}\:=\:\lfloor\frac{\mathrm{40}}{\mathrm{7}}\rfloor \\ $$$$\:\:\:\:\:\:\:\:\:{a}\:=\:\mathrm{5} \\ $$$$\:\: \\ $$$$\:\:{cba}\:−\:{abc}\:=\:\mathrm{655}\:−\:\mathrm{556} \\ $$$$\:\:=\:\mathrm{99} \\ $$

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