Puzzle for August 20, 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.6: A + B + B = E – B + B which becomes A + 2×B = E In eq.4, replace E with A + 2×B: A + 2×B – D = A In the above equation, subtract A from both sides, and add D to both sides: A + 2×B – D – A + D = A – A + D which makes 2×B = D
Hint #2
Add D to both sides of eq.2: C – D + D = E + D which becomes eq.2a) C = E + D In eq.5, replace C with E + D (from eq.2a), and replace A with E – D (from eq.4): B + E + D + D = E – D + E which becomes B + E + 2×D = 2×E – D In the above equation, subtract E from each side, and add D to both sides: B + E + 2×D – E + D = 2×E – D – E + D which simplifies to eq.5a) B + 3×D = E
Hint #3
In eq.5a, substitute (2×B) for D: B + 3×(2×B) = E which becomes B + 6×B = E which means 7×B = E
Hint #4
Substitute 7×B for E, and 2×B for D in eq.2a: C = 7×B + 2×B which makes C = 9×B
Hint #5
Substitute 7×B for E, and 2×B for D in eq.4: 7×B – 2×B = A which makes 5×B = A
Hint #6
Substitute 2×B for D, and 5×B for A in eq.3: 2×B + F = 5×B – B which becomes 2×B + F = 4×B Subtract 2×B from both sides: 2×B + F – 2×B = 4×B – 2×B which makes F = 2×B
Solution
Substitute 5×B for A, 9×B for C, 2×B for D and F, and 7×B for E in eq.1: 5×B + B + 9×B + 2×B + 7×B + 2×B = 26 which simplifies to 26×B = 26 Divide both sides of the equation above by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = 5×B = 5 × 1 = 5 C = 9×B = 9 × 1 = 9 D = F = 2×B = 2 × 1 = 2 E = 7×B = 7 × 1 = 7 and ABCDEF = 519272