Puzzle for December 18, 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 positive integer.
Scratchpad
Help Area
Hint #1
In eq.4, replace E with D + F (from eq.2): A + F = D + D + F which becomes A + F = 2×D + F Subtract F from each side of the equation above: A + F - F = 2×D + F - F which makes A = 2×D
Hint #2
In eq.1, replace A with 2×D: C = 2×D + D which makes C = 3×D
Hint #3
In eq.5, substitute 3×D for C: D - 3×D = 3×D - F which becomes -2×D = 3×D - F Add 2×D and F to both sides of the above equation: -2×D + 2×D + F = 3×D - F + 2×D + F which makes F = 5×D
Hint #4
Substitute 5×D for F in eq.2: E = D + 5×D which makes E = 6×D
Hint #5
Substitute 5×D for F, and 3×D for C in eq.3: 5×D = B + 3×D Subtract 3×D from each side of the above equation: 5×D - 3×D = B + 3×D - 3×D which makes 2×D = B
Solution
Substitute 6×D for E, 2×D for B, and 3×D for C in eq.6: 6×D = 2×D × 3×D which becomes 6×D = 6×D² Divide both sides of the above equation by 6×D: 6×D ÷ 6×D = 6×D² ÷ 6×D which makes 1 = D making A = B = 2×D = 2 × 1 = 2 C = 3×D = 3 × 1 = 3 E = 6×D = 6 × 1 = 6 F = 5×D = 5 × 1 = 5 and ABCDEF = 223165