Puzzle for July 3, 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.4, replace D with A + C (from eq.3): C + A + C = A + F which becomes 2×C + A = A + F Subtract A from each side of the above equation: 2×C + A - A = A + F - A which makes 2×C = F
Hint #2
In eq.2, replace F with 2×C: B = C + 2×C which makes B = 3×C
Hint #3
In eq.5, substitute 3×C for B: D - 3×C = 3×C - C which becomes D - 3×C = 2×C Add 3×C to both sides of the equation above: D - 3×C + 3×C = 2×C + 3×C which makes D = 5×C
Hint #4
Substitute 5×C for D in eq.3: 5×C = A + C Subtract C from each side of the equation above: 5×C - C = A + C - C which makes 4×C = A
Hint #5
Substitute 4×C for A, and 5×C for D in eq.6: 4×C + 5×C = C + E which becomes 9×C = C + E Subtract C from both sides of the above equation: 9×C - C = C + E - C which makes 8×C = E
Solution
Substitute 4×C for A, 3×C for B, 5×C for D, 8×C for E, and 2×C for F in eq.1: 4×C + 3×C + C + 5×C + 8×C + 2×C = 23 which simplifies to 23×C = 23 Divide both sides of the above equation by 23: 23×C ÷ 23 = 23 ÷ 23 which means C = 1 making A = 4×C = 4 × 1 = 4 B = 3×C = 3 × 1 = 3 D = 5×C = 5 × 1 = 5 E = 8×C = 8 × 1 = 8 F = 2×C = 2 × 1 = 2 and ABCDEF = 431582