Puzzle for June 6, 2019 ( )
Scratchpad
Find the 5-digit number ABCDE by solving the following equations:
A, B, C, D, and E each represent a one-digit non-negative integer.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add D to both sides of eq.4: E + D = A - D + D which becomes eq.4a) E + D = A Subtract E from both sides of eq.4a: E + D - E = A - E which becomes eq.4b) D = A - E
Hint #2
Subtract the left and right sides of eq.3 from the left and right sides of eq.4a, respectively: E + D - (D - E) = A - B which is the same as E + D - D + E = A - B which becomes 2×E = A - B In eq.2, replace A - B with 2×E: C = 2×E
Hint #3
eq.5 may be written as: B + 10×C + D = 10×A + B - D In the above equation, subtract B from both sides, and add D to both sides: B + 10×C + D - B + D = 10×A + B - D - B + D which simplifies to 10×C + 2×D = 10×A Substitute (2×E) for C, and (A - E) for D (from eq.4b): 10×(2×E) + 2×(A - E) = 10×A which is equivalent to 20×E + 2×A - 2×E = 10×A Subtract 2×A from both sides: 20×E + 2×A - 2×E - 2×A = 10×A - 2×A which becomes 18×E = 8×A Divide both sides by 8: 18×E ÷ 8 = 8×A ÷ 8 which means 2¼×E = A
Hint #4
Substitute 2¼×E for A in eq.4b: D = 2¼×E - E which makes D = 1¼×E
Hint #5
Substitute 2×E for C, and 2¼×E for A in eq.2: 2×E = 2¼×E - B Add B to both sides, and subtract 2×E from both sides: 2×E + B - 2×E = 2¼×E - B + B - 2×E which makes B = ¼×E
Solution
Substitute 2¼×E for A, ¼×E for B, 2×E for C, and 1¼×E for D in eq.1: 2¼×E + ¼×E + 2×E + 1¼×E + E = 27 which simplifies to 6¾×E = 27 Divide both sides by 6¾: 6¾×E ÷ 6¾ = 27 ÷ 6¾ which means E = 4 making A = 2¼×E = 2¼ × 4 = 9 B = ¼×E = ¼ × 4 = 1 C = 2×E = 2 × 4 = 8 D = 1¼×E = 1¼ × 4 = 5 and ABCDE = 91854