Puzzle for February 8, 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
eq.2 may be written as: E – F – D = B + D In the equation above, replace E – F with B + D + F (from eq.5): B + D + F – D = B + D which becomes B + F = B + D Subtract B from each side: B + F – B = B + D – B which makes F = D
Hint #2
In eq.6, replace F with D: D + D = A – D – D which becomes 2×D = A – 2×D Add 2×D to both sides of the above equation: 2×D + 2×D = A – 2×D + 2×D which makes 4×D = A
Hint #3
In eq.4, substitute 4×D for A: 4×D + D = B – D which becomes 5×D = B – D Add D to both sides of the equation above: 5×D + D = B – D + D which makes 6×D = B
Hint #4
Substitute D for F, and 6×D for B in eq.5: E – D = 6×D + D + D which becomes E – D = 8×D Add D to both sides of the above equation: E – D + D = 8×D + D which makes E = 9×D
Hint #5
Substitute 6×D for B, 4×D for A, and D for F in eq.3: 6×D + C = 4×D + D + D which becomes 6×D + C = 6×D Subtract 6×D from each side of the equation above: 6×D + C – 6×D = 6×D – 6×D which means C = 0
Solution
Substitute 4×D for A, 6×D for B, 0 for C, 9×D for E, and D for F in eq.1: 4×D + 6×D + 0 + D + 9×D + D = 21 which simplifies to 21×D = 21 Divide both sides of the above equation by 21: 21×D ÷ 21 = 21 ÷ 21 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 6×D = 6 × 1 = 6 E = 9×D = 9 × 1 = 9 F = D = 1 and ABCDEF = 460191