Puzzle for August 23, 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 A to both sides of eq.5: E - A + A = A - F + A which becomes eq.5a) E = 2×A - F Add F to both sides of eq.6: F - C + F = E - F + F which becomes eq.6a) 2×F - C = E
Hint #2
In eq.5a, replace E with 2×F - C (from eq.6a): 2×F - C = 2×A - F Add C and F to both sides of the above equation: 2×F - C + C + F = 2×A - F + C + F which becomes eq.5b) 3×F = 2×A + C
Hint #3
Add C to both sides of eq.3: C + F + C = A - C + C which becomes eq.3a) 2×C + F = A
Hint #4
In eq.5b, substitute (2×C + F) for A (from eq.3a): 3×F = 2×(2×C + F) + C which becomes 3×F = 4×C + 2×F + C which becomes 3×F = 5×C + 2×F Subtract 2×F from each side of the equation above: 3×F - 2×F = 5×C + 2×F - 2×F which makes F = 5×C
Hint #5
Substitute 5×C for F in eq.3a: 2×C + 5×C = A which makes 7×C = A
Hint #6
Substitute 7×C for A in eq.2: B - C = 7×C + C which becomes B - C = 8×C Add C to both sides of the equation above: B - C + C = 8×C + C which makes B = 9×C
Hint #7
Substitute (7×C) for A, and 5×C for F in eq.5a: E = 2×(7×C) - 5×C which becomes E = 14×C - 5×C which makes E = 9×C
Hint #8
Substitute 5×C for F, and 9×C for E in eq.4: D + 5×C = 9×C Subtract 5×C from each side of the equation above: D + 5×C - 5×C = 9×C - 5×C which makes D = 4×C
Solution
Substitute 7×C for A, 9×C for B and E, 4×C for D, and 5×C for F in eq.1: 7×C + 9×C + C + 4×C + 9×C + 5×C = 35 which simplifies to 35×C = 35 Divide both sides of the above equation by 35: 35×C ÷ 35 = 35 ÷ 35 which means C = 1 making A = 7×C = 7 × 1 = 7 B = E = 9×C = 9 × 1 = 9 D = 4×C = 4 × 1 = 4 F = 5×C = 5 × 1 = 5 and ABCDEF = 791495