Puzzle for February 13, 2020 ( )
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 E to both sides of eq.6: D – E + E = A + E + E which becomes D = A + 2×E In eq.4, replace D with A + 2×E: A + F = A + 2×E + E which becomes A + F = A + 3×E Subtract A from each side of the equation above: A + F – A = A + 3×E – A which makes F = 3×E
Hint #2
In eq.5, replace F with 3×E: E + 3×E = B which makes 4×E = B
Hint #3
In eq.2, substitute 3×E for F: D + 3×E = A + C + E Subtract E from both sides of the equation above: D + 3×E – E = A + C + E – E which becomes eq.2a) D + 2×E = A + C
Hint #4
Substitute 4×E for B in eq.3: eq.3a) 4×E + C = D
Hint #5
Substitute 4×E + C for D (from eq.3a) in eq.2a: 4×E + C + 2×E = A + C which becomes 6×E + C = A + C Subtract C from both sides of the above equation: 6×E + C – C = A + C – C which makes 6×E = A
Hint #6
Substitute 6×E for A in eq.6: D – E = 6×E + E which becomes D – E = 7×E Add E to each side of the equation above: D – E + E = 7×E + E which makes D = 8×E
Hint #7
Substitute 4×E for B, and 8×E for D in eq.3: 4×E + C = 8×E Subtract 4×E from both sides of the above equation: 4×E + C – 4×E = 8×E – 4×E which makes C = 4×E
Solution
Substitute 6×E for A, 4×E for B and C, 8×E for D, and 3×E for F in eq.1: 6×E + 4×E + 4×E + 8×E + E + 3×E = 26 which simplifies to 26×E = 26 Divide each side of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 6×E = 6 × 1 = 6 B = C = 4×E = 4 × 1 = 4 D = 8×E = 8 × 1 = 8 F = 3×E = 3 × 1 = 3 and ABCDEF = 644813