Puzzle for March 22, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* AB and DE are 2-digit numbers (not A×B or D×E).
Scratchpad
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Hint #1
Add B + D to both sides of eq.5: C - D + B + D = F - B + B + D which becomes C + B = F + D which may also be written as eq.5a) B + C = F + D In eq.2, replace B + C with F + D: F + D = E + F Subtract F from both sides of the above equation: F + D - F = E + F - F which makes D = E
Hint #2
eq.6 may be expressed as: 10×D + E - (10×A + B) = C + D which is equivalent to 10×D + E - 10×A - B = C + D Add (B - D) to both sides of the equation above: 10×D + E - 10×A - B + (B - D) = C + D + (B - D) which becomes 9×D + E - 10×A = C + B which may be written as 9×D + E - 10×A = B + C Substitute D for E, and F + D for B + C (from eq.5a): 9×D + D - 10×A = F + D Subtract D from each side: 9×D + D - 10×A - D = F + D - D which becomes eq.6a) 9×D - 10×A = F
Hint #3
Add A to each side of eq.4: D + E - A + A = F + A + A which becomes D + E = F + 2×A Substitute D for E, and (9×D - 10×A) for F (from eq.6a) in the equation above: D + D = (9×D - 10×A) + 2×A which becomes 2×D = 9×D - 8×A Add (8×A - 2×D) to each side: 2×D + (8×A - 2×D) = 9×D - 8×A + (8×A - 2×D) which simplifies to 8×A = 7×D Divide both sides by 8: 8×A ÷ 8 = 7×D ÷ 8 which means A = ⅞×D
Hint #4
In eq.6a, substitute (⅞×D) for A: 9×D - 10×(⅞×D) = F which becomes 9×D - 8¾×D = F which makes ¼×D = F
Hint #5
Substitute ⅞×D for A, and ¼×D for F in eq.3: ⅞×D = B + ¼×D Subtract ¼×D from both sides: ⅞×D - ¼×D = B + ¼×D - ¼×D which makes ⅝×D = B
Hint #6
Substitute ⅝×D for B, D for E, and ¼×D for F in eq.2: ⅝×D + C = D + ¼×D which becomes ⅝×D + C = 1¼×D Subtract ⅝×D from both sides: ⅝×D + C - ⅝×D = 1¼×D - ⅝×D which means C = ⅝×D
Solution
Substitute ⅞×D for A, ⅝×D for B and C, D for E, and ¼×D for F in eq.1: ⅞×D + ⅝×D + ⅝×D + D + D + ¼×D = 35 which simplifies to 4⅜×D = 35 Divide both sides by 4⅜: 4⅜×D ÷ 4⅜ = 35 ÷ 4⅜ which means D = 8 making A = ⅞×D = ⅞ × 8 = 7 B = C = ⅝×D = ⅝ × 8 = 5 E = D = 8 F = ¼×D = ¼ × 8 = 2 and ABCDEF = 755882