Puzzle for July 10, 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
In eq.5, replace C with B + D (from eq.3): D + F = B + B + D which becomes D + F = 2×B + D Subtract D from each side of the equation above: D + F - D = 2×B + D - D which makes F = 2×B
Hint #2
In eq.6, replace E with C + F (from eq.4): C + F + F = A + C which becomes C + 2×F = A + C Subtract C from each side of the equation above: C + 2×F - C = A + C - C which makes eq.6a) 2×F = A
Hint #3
In eq.6a, substitute (2×B) for F: 2×(2×B) = A which makes 4×B = A
Hint #4
Substitute 4×B for A in eq.2: D = 4×B + B which makes D = 5×B
Hint #5
Substitute 5×B for D, and 2×B for F in eq.5: 5×B + 2×B = B + C which becomes 7×B = B + C Subtract B from both sides of the equation above: 7×B - B = B + C - B which makes 6×B = C
Hint #6
Substitute 6×B for C, and 2×B for F in eq.4: E = 6×B + 2×B which makes E = 8×B
Solution
Substitute 4×B for A, 6×B for C, 5×B for D, 8×B for E, and 2×B for F in eq.1: 4×B + B + 6×B + 5×B + 8×B + 2×B = 26 which simplifies to 26×B = 26 Divide both sides of the above equation by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = 4×B = 4 × 1 = 4 C = 6×B = 6 × 1 = 6 D = 5×B = 5 × 1 = 5 E = 8×B = 8 × 1 = 8 F = 2×B = 2 × 1 = 2 and ABCDEF = 416582