Puzzle for May 22, 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 to both sides of eq.6: B + C + B = A - B + B which becomes eq.6a) 2×B + C = A
Hint #2
In eq.4, replace A with 2×B + C (from eq.6a): 2×B + C - C = C - B which becomes 2×B = C - B Add B to both sides of the above equation: 2×B + B = C - B + B which makes 3×B = C
Hint #3
In eq.6a, substitute 3×B for C: 2×B + 3×B = A which makes 5×B = A
Hint #4
Substitute 5×B for A in eq.3: F = 5×B + B which makes F = 6×B
Hint #5
Substitute 5×B for A, and 3×B for C in eq.2: E = 5×B + 3×B which makes E = 8×B
Hint #6
Substitute 3×B for C, and 5×B for A in eq.5: 3×B - D = D - 5×B Add D and 5×B to both sides of the equation above: 3×B - D + D + 5×B = D - 5×B + D + 5×B which makes 8×B = 2×D Divide both sides by 2: 8×B ÷ 2 = 2×D ÷ 2 which makes 4×B = D
Solution
Substitute 5×B for A, 3×B for C, 4×B for D, 8×B for E, and 6×B for F in eq.0: 5×B + B + 3×B + 4×B + 8×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 = 5×B = 5 × 1 = 5 C = 3×B = 3 × 1 = 3 D = 4×B = 4 × 1 = 4 E = 8×B = 8 × 1 = 8 F = 6×B = 6 × 1 = 6 and ABCDEF = 513486