Puzzle for January 17, 2022 ( )
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.6, replace C with D + F (from eq.3): D + F + D = A + F which becomes 2×D + F = A + F Subtract F from each side of the equation above: 2×D + F – F = A + F – F which makes 2×D = A
Hint #2
In eq.2, replace A with 2×D: E = 2×D + D which makes E = 3×D
Hint #3
In eq.5, replace A with 2×D, and E with 3×D: 2×D + 3×D = F – 3×D which becomes 5×D = F – 3×D Add 3×D to both sides of the above equation: 5×D + 3×D = F – 3×D + 3×D which makes 8×D = F
Hint #4
Substitute 8×D for F in eq.3: C = D + 8×D which makes C = 9×D
Hint #5
Substitute 8×D for F in eq.4: 8×D = B + D Subtract D from each side of the above equation: 8×D – D = B + D – D which makes 7×D = B
Solution
In eq.1, substitute 2×D for A, 7×D for B, 9×D for C, 3×D for E, and 8×D for F: 2×D + 7×D + 9×D + D + 3×D + 8×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 = 2×D = 2 × 1 = 2 B = 7×D = 7 × 1 = 7 C = 9×D = 9 × 1 = 9 E = 3×D = 3 × 1 = 3 F = 8×D = 8 × 1 = 8 and ABCDEF = 279138