Puzzle for October 21, 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 B to both sides of eq.3: C + E + B = A – B + B which becomes C + E + B = A which may be written as B + C + E = A In the equation above, replace B + C with E (from eq.2): E + E = A which makes 2×E = A
Hint #2
In eq.4, replace A with 2×E: D = 2×E + E which makes D = 3×E
Hint #3
In eq.5, substitute 2×E for A: E = F – 2×E Add 2×E to each side of the above equation: E + 2×E = F – 2×E + 2×E which makes 3×E = F
Hint #4
Substitute 3×E for F in eq.6: 3×E – C = C + E In the above equation, add C to both sides, and subtract E from each side: 3×E – C + C – E = C + E + C – E which means 2×E = 2×C Divide both sides by 2: 2×E ÷ 2 = 2×C ÷ 2 which makes E = C
Hint #5
Substitute C for E in eq.2: B + C = C Subtract C from both sides: B + C – C = C – C which means B = 0
Solution
Substitute 2×E for A, 0 for B, E for C, and 3×E for D and F in eq.1: 2×E + 0 + E + 3×E + E + 3×E = 30 which simplifies to 10×E = 30 Divide both sides of the equation above by 10: 10×E ÷ 10 = 30 ÷ 10 which means E = 3 making A = 2×E = 2 × 3 = 6 C = E = 3 D = F = 3×E = 3 × 3 = 9 and ABCDEF = 603939