Puzzle for May 24, 2022 ( )
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.3, replace B with A + E (from eq.2): C = A + E + E which becomes eq.3a) C = A + 2×E
Hint #2
In eq.4, replace C with A + 2×E (from eq.3a): F = A + 2×E + E which becomes eq.4a) F = A + 3×E
Hint #3
In eq.5, substitute A + 3×E for F (from eq.4a): D = A + 3×E + E which becomes eq.5a) D = A + 4×E
Hint #4
Substitute A + 4×E for D (from eq.5a) into eq.6: A + B = A + 4×E + E which becomes A + B = A + 5×E Subtract A from each side of the above equation: A + B – A = A + 5×E – A which makes B = 5×E
Hint #5
Substitute 5×E for B in eq.2: 5×E = A + E Subtract E from each side of the equation above: 5×E – E = A + E – E which makes 4×E = A
Hint #6
Substitute 4×E for A in eq.3a: C = 4×E + 2×E which makes C = 6×E
Hint #7
Substitute 4×E for A in eq.5a: D = 4×E + 4×E which makes D = 8×E
Hint #8
Substitute 4×E for A in eq.4a: F = 4×E + 3×E which makes F = 7×E
Solution
Substitute 4×E for A, 5×E for B, 6×E for C, 8×E for D, and 7×E for F in eq.1: 4×E + 5×E + 6×E + 8×E + E + 7×E = 31 which simplifies to 31×E = 31 Divide both sides of the above equation by 31: 31×E ÷ 31 = 31 ÷ 31 which means E = 1 making A = 4×E = 4 × 1 = 4 B = 5×E = 5 × 1 = 5 C = 6×E = 6 × 1 = 6 D = 8×E = 8 × 1 = 8 F = 7×E = 7 × 1 = 7 and ABCDEF = 456817