Puzzle for September 27, 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
Help Area
Hint #1
In eq.4, replace A with C + D (from eq.3): B + C = C + D + D which becomes B + C = C + 2×D Subtract C from each side of the equation above: B + C – C = C + 2×D – C which makes B = 2×D
Hint #2
In eq.5, replace B with 2×D: E – D = 2×D + D which becomes E – D = 3×D Add D to both sides of the above equation: E – D + D = 3×D + D which makes E = 4×D
Hint #3
In eq.2, substitute 4×D for E: F = D + 4×D which makes F = 5×D
Hint #4
Substitute 4×D for E in eq.6: D + 4×D = A – 4×D which becomes 5×D = A – 4×D Add 4×D to both sides of the above equation: 5×D + 4×D = A – 4×D + 4×D which makes 9×D = A
Hint #5
Substitute 9×D for A in eq.3: 9×D = C + D Subtract D from each side of the above equation: 9×D – D = C + D – D which makes 8×D = C
Solution
Substitute 9×D for A, 2×D for B, 8×D for C, 4×D for E, and 5×D for F in eq.1: 9×D + 2×D + 8×D + D + 4×D + 5×D = 29 which simplifies to 29×D = 29 Divide both sides of the above equation by 29: 29×D ÷ 29 = 29 ÷ 29 which means D = 1 making A = 9×D = 9 × 1 = 9 B = 2×D = 2 × 1 = 2 C = 8×D = 8 × 1 = 8 E = 4×D = 4 × 1 = 4 F = 5×D = 5 × 1 = 5 and ABCDEF = 928145