Puzzle for November 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 B + C with D + E (from eq.2): A – D + E = D + E + D which becomes A – D + E = 2×D + E In the above equation, subtract E from both sides, and add D to both sides: A – D + E – E + D = 2×D + E – E + D which makes A = 3×D
Hint #2
In eq.4, replace A with 3×D: B = 3×D + D which makes B = 4×D
Hint #3
In eq.3, replace B with 4×D: C = 4×D + D which makes C = 5×D
Hint #4
In eq.2, substitute 4×D for B, and 5×D for C: D + E = 4×D + 5×D which becomes D + E = 9×D Subtract D from each side of the above equation: D + E – D = 9×D – D which makes E = 8×D
Hint #5
Substitute 8×D for E in eq.5: F = D + 8×D which makes F = 9×D
Solution
Substitute 3×D for A, 4×D for B, 5×D for C, 8×D for E, and 9×D for F in eq.1: 3×D + 4×D + 5×D + D + 8×D + 9×D = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 4×D = 4 × 1 = 4 C = 5×D = 5 × 1 = 5 E = 8×D = 8 × 1 = 8 F = 9×D = 9 × 1 = 9 and ABCDEF = 345189