Puzzle for September 27, 2022 ( )
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
In eq.5, replace A + B with D (from eq.3): C = D + D which makes eq.5a) C = 2×D
Hint #2
Add A to both sides of eq.6: E - A + A = A - B + A which becomes eq.6a) E = 2×A - B
Hint #3
In eq.4, replace E with 2×A - B (from eq.6a), and D with A + B (from eq.3): 2×A - B = B + A + B which becomes 2×A - B = 2×B + A In the above equation, add B to both sides, and subtract A from both sides: 2×A - B + B - A = 2×B + A + B - A which makes A = 3×B
Hint #4
In eq.6a, substitute (3×B) for A: E = 2×(3×B) - B which becomes E = 6×B - B which makes E = 5×B
Hint #5
Substitute 5×B for E in eq.4: 5×B = B + D Subtract B from each side of the above equation: 5×B - B = B + D - B which makes 4×B = D
Hint #6
Substitute (4×B) for D in eq.5a: C = 2×(4×B) which makes C = 8×B
Hint #7
Substitute 8×B for C, 4×B for D, and 5×B for E in eq.2: 8×B + 4×B = B + 5×B + F which becomes 12×B = 6×B + F Subtract 6×B from each side of the equation above: 12×B - 6×B = 6×B + F - 6×B which makes 6×B = F
Solution
Substitute 3×B for A, 8×B for C, 4×B for D, 5×B for E, and 6×B for F in eq.1: 3×B + B + 8×B + 4×B + 5×B + 6×B = 27 which simplifies to 27×B = 27 Divide both sides of the above equation by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 8×B = 8 × 1 = 8 D = 4×B = 4 × 1 = 4 E = 5×B = 5 × 1 = 5 F = 6×B = 6 × 1 = 6 and ABCDEF = 318456