Puzzle for September 23, 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 E to both sides of eq.2: F – E + E = C + E + E which becomes F = C + 2×E In eq.3, replace F with C + 2×E: B + C = C + 2×E Subtract C from both sides of the above equation: B + C – C = C + 2×E – C which becomes B = 2×E
Hint #2
In eq.4, replace B with 2×E: D = 2×E + E D = 3×E
Hint #3
In eq.5, substitute 3×E for D: C – E = 3×E + E C – E = 4×E Add E to both sides of the above equation: C – E + E = 4×E + E C = 5×E
Hint #4
Substitute 5×E for C, and 3×E for D In eq.6: A = 5×E + 3×E A = 8×E
Hint #5
Substitute 2×E for B, and 5×E for C in eq.3: 2×E + 5×E = F 7×E = F
Solution
Substitute 8×E for A, 2×E for B, 5×E for C, 3×E for D, and 7×E for F in eq.1: 8×E + 2×E + 5×E + 3×E + E + 7×E = 26 which simplifies to 26×E = 26 Divide both sides of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 2×E = 2 × 1 = 2 C = 5×E = 5 × 1 = 5 D = 3×E = 3 × 1 = 3 F = 7×E = 7 × 1 = 7 and ABCDEF = 825317