Puzzle for July 12, 2021 ( )
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): B + E = C + E + C which becomes B + E = 2×C + E Subtract E from both sides of the above equation: B + E – E = 2×C + E – E which makes B = 2×C
Hint #2
In eq.6, replace B with 2×C: D – 2×C = 2×C + C which becomes D – 2×C = 3×C Add 2×C to both sides of the above equation: D – 2×C + 2×C = 3×C + 2×C which makes D = 5×C
Hint #3
In eq.2, substitute 2×C for B, and 5×C for D: F = 2×C + 5×C which makes F = 7×C
Hint #4
In eq.5, substitute 2×C for B, and 5×C for D: E = 2×C + C + 5×C which makes E = 8×C
Hint #5
Substitute 8×C for E in eq.3: C + 8×C = A which makes 9×C = A
Solution
Substitute 9×C for A, 2×C for B, 5×C for D, 8×C for E, and 7×C for F in eq.1: 9×C + 2×C + C + 5×C + 8×C + 7×C = 32 which simplifies to 32×C = 32 Divide both sides of the equation above by 32: 32×C ÷ 32 = 32 ÷ 32 which means C = 1 making A = 9×C = 9 × 1 = 9 B = 2×C = 2 × 1 = 2 D = 5×C = 5 × 1 = 5 E = 8×C = 8 × 1 = 8 F = 7×C = 7 × 1 = 7 and ABCDEF = 921587