Puzzle for July 24, 2023 ( )
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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.
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Hint #1
Add B to both sides of eq.6: E - B + B = B + D + B which becomes eq.6a) E = 2×B + D
Hint #2
In eq.5, substitute (2×B + D) for E (from eq.6a): B - D = D - (2×B + D) which becomes B - D = D - 2×B - D which becomes B - D = -2×B Add D and 2×B to both sides of the above equation: B - D + D + 2×B = -2×B + D + 2×B which makes 3×B = D
Hint #3
In eq.6a, substitute 3×B for D: E = 2×B + 3×B which makes E = 5×B
Hint #4
In eq.4, substitute 3×B for D, and 5×B for E: F = 3×B + 5×B which makes F = 8×B
Hint #5
Substitute 5×B for E in eq.3: A = B + 5×B which makes A = 6×B
Hint #6
Substitute 8×B for F in eq.2: C = B + 8×B which makes C = 9×B
Solution
Substitute 6×B for A, 9×B for C, 3×B for D, 5×B for E, and 8×B for F in eq.1: 6×B + B + 9×B + 3×B + 5×B + 8×B = 32 which simplifies to 32×B = 32 Divide both sides of the above equation by 32: 32×B ÷ 32 = 32 ÷ 32 which means B = 1 making A = 6×B = 6 × 1 = 6 C = 9×B = 9 × 1 = 9 D = 3×B = 3 × 1 = 3 E = 5×B = 5 × 1 = 5 F = 8×B = 8 × 1 = 8 and ABCDEF = 619358