Puzzle for May 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
In eq.4, replace D with C + E (from eq.6): C + F = A + C + E Subtract C from each side of the equation above: C + F – C = A + C + E – C which becomes eq.4a) F = A + E
Hint #2
In eq.4a, replace A + E with B + F (from eq.3): F = B + F Subtract F from both sides of the equation above: F – F = B + F – F which makes 0 = B
Hint #3
In eq.2, substitute D with C + E (from eq.6), and 0 for B: C + E + E = A + 0 + C which becomes C + 2×E = A + C Subtract C from both sides of the equation above: C + 2×E – C = A + C – C which becomes 2×E = A
Hint #4
Substitute 2×E for A in eq.4a: F = 2×E + E which makes F = 3×E
Hint #5
Add D to both sides of eq.5: A – D + F – D = C + D + E + D which becomes A + F = C + E + 2×D Substitute D for C + E (from eq.6) in the above equation: A + F = D + 2×D which becomes eq.5a) A + F = 3×D
Hint #6
Substitute 2×E for A, and 3×E for F in eq.5a: 2×E + 3×E = 3×D which makes 5×E = 3×D Divide both sides of the above equation by 3: 5×E ÷ 3 = 3×D ÷ 3 which makes 1⅔×E = D
Hint #7
Substitute 1⅔×E for D in eq.6: 1⅔×E = C + E Subtract E from each side of the above equation: 1⅔×E – E = C + E – E which makes ⅔×E = C
Solution
Substitute 2×E for A, 0 for B, ⅔×E for C, 1⅔×E for D, and 3×E for F in eq.1: 2×E + 0 + ⅔×E + 1⅔×E + E + 3×E = 25 which simplifies to 8⅓×E = 25 Divide both sides of the equation above by 8⅓: 8⅓×E ÷ 8⅓ = 25 ÷ 8⅓ which means E = 3 making A = 2×E = 2 × 3 = 6 C = ⅔×E = ⅔ × 3 = 2 D = 1⅔×E = 1⅔ × 3 = 5 F = 3×E = 3 × 3 = 9 and ABCDEF = 602539