Puzzle for October 9, 2023 ( )
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
Help Area
Hint #1
In eq.4, replace C with D + F (from eq.2): B + F = D + F + D which becomes B + F = 2×D + F Subtract F from each side of the equation above: B + F - F = 2×D + F - F which makes B = 2×D
Hint #2
In eq.5, replace B with 2×D: A - D = 2×D + D which becomes A - D = 3×D Add D to each side of the above equation: A - D + D = 3×D + D which makes A = 4×D
Hint #3
In eq.6, substitute 4×D for A, and 2×D for B: D + E = 4×D + 2×D which becomes D + E = 6×D Subtract D from both sides of the above equation: D + E - D = 6×D - D which makes E = 5×D
Hint #4
In eq.3, substitute 4×D for A, and 2×D for B: F = 4×D + 2×D + D which makes F = 7×D
Hint #5
Substitute 7×D for F in eq.2: C = D + 7×D which makes C = 8×D
Solution
Substitute 4×D for A, 2×D for B, 8×D for C, 5×D for E, and 7×D for F in eq.1: 4×D + 2×D + 8×D + D + 5×D + 7×D = 27 which simplifies to 27×D = 27 Divide both sides of the above equation by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 2×D = 2 × 1 = 2 C = 8×D = 8 × 1 = 8 E = 5×D = 5 × 1 = 5 F = 7×D = 7 × 1 = 7 and ABCDEF = 428157