Puzzle for March 13, 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 EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Add D + F to both sides of eq.4: E - F + D + F = A - D + D + F which becomes E + D = A + F In the equation above, replace A + F with C + D (from eq.2): E + D = C + D Subtract D from both sides: E + D - D = C + D - D which makes E = C
Hint #2
In eq.3, replace E with C: B + C = D + C Subtract C from each side of the above equation: B + C - C = D + C - C which makes B = D
Hint #3
eq.6 may be written as: 10×A + B + D + F = 10×E + F Subtract F from both sides of the above equation: 10×A + B + D + F - F = 10×E + F - F which becomes 10×A + B + D = 10×E Substitute B for D, and C for E: 10×A + B + B = 10×C which becomes eq.6a) 10×A + 2×B = 10×C
Hint #4
In eq.5, substitute B for D, and C for E: A + B + B = C + C which becomes A + 2×B = 2×C Subtract A from each side of the equation above: A + 2×B - A = 2×C - A which becomes eq.5a) 2×B = 2×C - A
Hint #5
In eq.6a, substitute 2×C - A for 2×B (from eq.5a): 10×A + 2×C - A = 10×C Subtract 2×C from each side: 10×A + 2×C - A - 2×C = 10×C - 2×C which simplifies to 9×A = 8×C Divide both sides by 8: 9×A ÷ 8 = 8×C ÷ 8 which means 1⅛×A = C which also means E = C = 1⅛×A
Hint #6
Substitute B for D, and 1⅛×A for both C and E in eq.5: A + B + B = 1⅛×A + 1⅛×A which becomes A + 2×B = 2¼×A Subtract A from both sides: A + 2×B - A = 2¼×A - A which makes 2×B = 1¼×A Divide both sides by 2: 2×B ÷ 2 = 1¼×A ÷ 2 which means B = ⅝×A which also means D = B = ⅝×A
Hint #7
Substitute 1⅛×A for C, and ⅝×A for D in eq.2: 1⅛×A + ⅝×A = A + F which becomes 1¾×A = A + F Subtract A from both sides: 1¾×A - A = A + F - A which makes ¾×A = F
Solution
Substitute ⅝×A for B and D, 1⅛×A for C and E, and ¾×A for F in eq.1: A + ⅝×A + 1⅛×A + ⅝×A + 1⅛×A + ¾×A = 42 which simplifies to 5¼×A = 42 Divide both sides by 5¼: 5¼×A ÷ 5¼ = 42 ÷ 5¼ which makes A = 8 making B = D = ⅝×A = ⅝ × 8 = 5 C = E = 1⅛×A = 1⅛ × 8 = 9 F = ¾×A = ¾ × 8 = 6 and ABCDEF = 859596