Puzzle for December 8, 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.3, replace F with A – E (from eq.6): E + A – E = A + D which becomes A = A + D Subtract A from each side of the equation above: A – A = A + D – A which means 0 = D
Hint #2
Add E to both sides of eq.4: C + F + E = B + D – E + E which becomes C + F + E = B + D which is the same as C + E + F = B + D In the above equation, replace E + F with A + D (from eq.3): C + A + D = B + D Subtract D from each side: C + A + D – D = B + D – D which becomes eq.4a) C + A = B
Hint #3
In eq.5, substitute C + A for B (from eq.4a): A + E = C + A – E In the equation above, subtract A from each side, and add E to each side: A + E – A + E = C + A – E – A + E which makes 2×E = C
Hint #4
In eq.2, substitute C + A for B (from eq.4a): C + A + C = A – C + F which becomes A + 2×C = A – C + F In the equation above, subtract A from both sides, and add C to both sides: A + 2×C – A + C = A – C + F – A + C which makes eq.2a) 3×C = F
Hint #5
Substitute 2×E for C in eq.2a: 3×(2×E) = F which makes 6×E = F
Hint #6
Substitute 6×E for F in eq.6: 6×E = A – E Add E to both sides of the equation above: 6×E + E = A – E + E which makes 7×E = A
Hint #7
Substitute 2×E for C, and 7×E for A in eq.4a: 2×E + 7×E = B which makes 9×E = B
Solution
Substitute 7×E for A, 9×E for B, 2×E for C, 0 for D, and 6×E for F in eq.1: 7×E + 9×E + 2×E + 0 + E + 6×E = 25 which simplifies to 25×E = 25 Divide both sides of the equation above by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 7×E = 7 × 1 = 7 B = 9×E = 9 × 1 = 9 C = 2×E = 2 × 1 = 2 F = 6×E = 6 × 1 = 6 and ABCDEF = 792016