Puzzle for August 22, 2022 ( )
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 C + E (from eq.3): E + F = C + E + C which becomes E + F = 2×C + E Subtract E from each side of the equation above: E + F – E = 2×C + E – E which makes F = 2×C
Hint #2
In eq.2, replace F with 2×C: D = C + 2×C which makes D = 3×C
Hint #3
In eq.5, replace F with 2×C: C + 2×C = A – 2×C which becomes 3×C = A – 2×C Add 2×C to both sides of the above equation: 3×C + 2×C = A – 2×C + 2×C which makes 5×C = A
Hint #4
In eq.3, substitute 5×C for A: 5×C = C + E Subtract C from each side of the above equation: 5×C – C = C + E – C which makes 4×C = E
Hint #5
Substitute 2×C for F, 3×C for D, and 4×C for E in eq.6: B + 2×C = 3×C + 4×C which becomes B + 2×C = 7×C Subtract 2×C from both sides of the equation above: B + 2×C – 2×C = 7×C – 2×C which makes B = 5×C
Solution
Substitute 5×C for A and B, 3×C for D, 4×C for E, and 2×C for F in eq.1: 5×C + 5×C + C + 3×C + 4×C + 2×C = 20 which simplifies to 20×C = 20 Divide both sides of the above equation by 20: 20×C ÷ 20 = 20 ÷ 20 which means C = 1 making A = B = 5×C = 5 × 1 = 5 D = 3×C = 3 × 1 = 3 E = 4×C = 4 × 1 = 4 F = 2×C = 2 × 1 = 2 and ABCDEF = 551342