Puzzle for March 23, 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.6, replace F with A + B (from eq.2): A + C = B + A + B which becomes A + C = A + 2×B Subtract A from each side of the equation above: A + C – A = A + 2×B – A which makes eq.6a) C = 2×B
Hint #2
In eq.5, replace C with 2×B (from eq.6a): 2×B – E = B + E In the above equation, add E to both sides, and subtract B from both sides: 2×B – E + E – B = B + E + E – B which makes B = 2×E
Hint #3
In eq.6a, substitute (2×E) for B: C = 2×(2×E) which makes C = 4×E
Hint #4
Substitute 2×E for B, and 4×E for C in eq.4: 2×E + 4×E = A – E which becomes 6×E = A – E Add E to each side of the equation above: 6×E + E = A – E + E which makes 7×E = A
Hint #5
Substitute 7×E for A in eq.3: D = 7×E + E which makes D = 8×E
Hint #6
Substitute 7×E for A, and 2×E for B in eq.2: F = 7×E + 2×E which makes F = 9×E
Solution
Substitute 7×E for A, 2×E for B, 4×E for C, 8×E for D, and 9×E for F in eq.1: 7×E + 2×E + 4×E + 8×E + E + 9×E = 31 which simplifies to 31×E = 31 Divide both sides of the above equation by 31: 31×E ÷ 31 = 31 ÷ 31 which means E = 1 making A = 7×E = 7 × 1 = 7 B = 2×E = 2 × 1 = 2 C = 4×E = 4 × 1 = 4 D = 8×E = 8 × 1 = 8 F = 9×E = 9 × 1 = 9 and ABCDEF = 724819