Puzzle for January 16, 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 A with D – E (from eq.3): B = D – E + E which makes B = D
Hint #2
Add E to both sides of eq.5: C + E + E = B – E + E which makes C + 2×E = B which also makes eq.5a) D = B = C + 2×E
Hint #3
In eq.2, replace B with C + 2×E (from eq.5a): F = C + 2×E + E which becomes eq.2a) F = C + 3×E
Hint #4
In eq.4, replace B with C + 2×E (from eq.5a): C + 2×E = A + E Subtract E from both sides of the equation above: C + 2×E – E = A + E – E which becomes eq.4a) C + E = A
Hint #5
Substitute C + E for A (from eq.4a), C + 2×E for B (from eq.5a), and C + 3×E for F (from eq.2a) in eq.6: C + E + C – E = C + 2×E – C + C + 3×E which becomes 2×C = C + 5×E Subtract C from each side of the above equation: 2×C – C = C + 5×E – C which makes C = 5×E
Hint #6
Substitute 5×E for C in eq.5a: D = B = 5×E + 2×E which makes D = B = 7×E
Hint #7
Substitute 5×E for C in eq.2a: F = 5×E + 3×E F = 8×E
Hint #8
Substitute 5×E for C in eq.4a: 5×E + E = A which makes 6×E = A
Solution
Substitute 6×E for A, 7×E for B and D, 5×E for C, and 8×E for F in eq.1: 6×E + 7×E + 5×E + 7×E + E + 8×E = 34 which simplifies to 34×E = 34 Divide both sides of the above equation by 34: 34×E ÷ 34 = 34 ÷ 34 which means E = 1 making A = 6×E = 6×1 = 6 B = D = 7×E = 7×1 = 7 C = 5×E = 5×1 = 5 F = 8×E = 8×1 = 8 and ABCDEF = 675718