Puzzle for June 21, 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.6, replace A with D + F (from eq.3): C + F = D + F + D which becomes C + F = 2×D + F Subtract F from both sides of the above equation: C + F – F = 2×D + F – F which makes C = 2×D
Hint #2
In eq.4, replace C with 2×D: E = 2×D + D which makes E = 3×D
Hint #3
In eq.5, substitute 3×D for E: F = D + 3×D which makes F = 4×D
Hint #4
Substitute 4×D for F into eq.3: D + 4×D = A which makes 5×D = A
Hint #5
Substitute 3×D for E, and 4×D for F in eq.2: B = 3×D + 4×D which makes B = 7×D
Solution
Substitute 5×D for A, 7×D for B, 2×D for C, 3×D for E, and 4×D for F in eq.1: 5×D + 7×D + 2×D + D + 3×D + 4×D = 22 which simplifies to 22×D = 22 Divide both sides of the above equation by 22: 22×D ÷ 22 = 22 ÷ 22 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 7×D = 7 × 1 = 7 C = 2×D = 2 × 1 = 2 E = 3×D = 3 × 1 = 3 F = 4×D = 4 × 1 = 4 and ABCDEF = 572134