Puzzle for February 8, 2023 ( )
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.
* EF and CD are 2-digit numbers (not E×F or C×D).
Scratchpad
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Hint #1
Add F and C to both sides of eq.3: D - F + F + C = A - C + F + C which becomes D + C = A + F which may be written as eq.3a) C + D = A + F
Hint #2
In eq.2, replace C + D with A + F (from eq.3a): A + F = E + F Subtract F from each side of the equation above: A + F - F = E + F - F which makes A = E
Hint #3
In eq.6, replace E with A: C ÷ A = F ÷ A Multiply both sides of the above equation by A: A × (C ÷ A) = A × (F ÷ A) which makes C = F
Hint #4
eq.4 may be written as: 10×E + F = A + 10×C + D which is equivalent to eq.4a) 10×E + F = A + 9×C + C + D
Hint #5
In eq.4a, substitute E for A, and E + F for C + D (from eq.2): 10×E + F = E + 9×C + E + F which becomes 10×E + F = 2×E + 9×C + F Subtract F and 2×E from each side of the above equation: 10×E + F - F - 2×E = 2×E + 9×C + F - F - 2×E which simplifies to 8×E = 9×C Divide both sides by 8: 8×E ÷ 8 = 9×C ÷ 8 which makes E = 1⅛×C and also makes A = E = 1⅛×C
Hint #6
Substitute 1⅛×C for A, and C for F in eq.3a: C + D = 1⅛×C + C Subtract C from each side of the equation above: C + D - C = 1⅛×C + C - C which makes D = 1⅛×C
Hint #7
eq.5 may be written as: F = (A + B + D + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × F = 4 × (A + B + D + E) ÷ 4 which becomes eq.5a) 4×F = A + B + D + E
Hint #8
Substitute C for F, and 1⅛×C for A and D and E in eq.5a: 4×C = 1⅛×C + B + 1⅛×C + 1⅛×C which becomes 4×C = 3⅜×C + B Subtract 3⅜×C from both sides of the above equation: 4×C - 3⅜×C = 3⅜×C + B - 3⅜×C which makes ⅝×C = B
Solution
Substitute 1⅛×C for A and D and E, ⅝×C for B, and C for F in eq.1: 1⅛×C + ⅝×C + C + 1⅛×C + 1⅛×C + C = 48 which simplifies to 6×C = 48 Divide both sides of the above equation by 6: 6×C ÷ 6 = 48 ÷ 6 which means C = 8 making A = D = E = 1⅛×C = 1⅛ × 8 = 9 B = ⅝×C = ⅝ × 8 = 5 F = C = 8 and ABCDEF = 958998