Puzzle for March 9, 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.
Scratchpad
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Hint #1
Add B and D to both sides of eq.5: D - B + B + D = A - D + B + D which becomes eq.5a) 2×D = A + B In eq.2, replace A + B with 2×D (from eq.5a): eq.2a) F = 2×D
Hint #2
Add B to both sides of eq.4: A - B + B = B + E + B which becomes eq.4a) A = 2×B + E
Hint #3
In eq.6, replace A with 2×B + E (from eq.4a): 2×B + E + D = B + C + E Subtract E and B from each side of the equation above: 2×B + E + D - E - B = B + C + E - E - B which simplifies to eq.6a) B + D = C
Hint #4
In eq.3, substitute 2×D for F: B + 2×D = D + E Subtract D from both sides of the equation above: B + 2×D - D = D + E - D which becomes eq.3a) B + D = E
Hint #5
Substitute C for B + D (from eq.6a) into eq.3a: C = E
Hint #6
Substitute B + D for E (from eq.3a) in eq.4a: A = 2×B + B + D which becomes eq.4b) A = 3×B + D
Hint #7
Substitute 3×B + D for A (from eq.4b) in eq.5a: 2×D = 3×B + D + B which becomes 2×D = 4×B + D Subtract D from each side of the above equation: 2×D - D = 4×B + D - D which makes D = 4×B
Hint #8
Substitute (4×B) for D in eq.2a: F = 2×(4×B) which makes F = 8×B
Hint #9
Substitute 4×B for D in eq.6a: B + 4×B = C which makes 5×B = C and also makes 5×B = C = E
Hint #10
Substitute 4×B for D in eq.4b: A = 3×B + 4×B which makes A = 7×B
Solution
Substitute 7×B for A, 5×B for C and E, 4×B for D, and 8×B for F in eq.1: 7×B + B + 5×B + 4×B + 5×B + 8×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = 7×B = 7 × 1 = 7 C = E = 5×B = 5 × 1 = 5 D = 4×B = 4 × 1 = 4 F = 8×B = 8 × 1 = 8 and ABCDEF = 715458