Puzzle for August 12, 2019 ( )
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.5: D + E + E = B + C – E + E which becomes D + 2×E = B + C In eq.6, replace B + C with D + 2×E: D + 2×E + E = D + F Subtract D from both sides of the equation above: D + 2×E + E – D = D + F – D which makes 3×E = F
Hint #2
In eq.2, replace F with 3×E: E + 3×E = A which makes 4×E = A
Hint #3
In eq.4, substitute 4×E for A: 4×E – C = C – E Add both C and E to each side of the equation above: 4×E – C + C + E = C – E + C + E which becomes 5×E = 2×C Divide both sides by 2: 5×E ÷ 2 = 2×C ÷ 2 which makes 2½×E = C
Hint #4
Substitute 2½×E for C in eq.3: 2½×E + E = B which makes 3½×E = B
Hint #5
Substitute 3½×E for B, and 2½×E for C in eq.5: D + E = 3½×E + 2½×E – E which becomes D + E = 5×E Subtract E from both sides of the equation above: D + E – E = 5×E – E which makes D = 4×E
Solution
Substitute 4×E for A and D, 3½×E for B, 2½×E for C, and 3×E for F in eq.1: 4×E + 3½×E + 2½×E + 4×E + E + 3×E = 36 which simplifies to 18×E = 36 Divide both sides by 18: 18×E ÷ 18 = 36 ÷ 18 which means E = 2 making A = D = 4×E = 4 × 2 = 8 B = 3½×E = 3½ × 2 = 7 C = 2½×E = 2½ × 2 = 5 F = 3×E = 3 × 2 = 6 and ABCDEF = 875826