Puzzle for February 20, 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.5, replace F with D + E (from eq.3): E - D = D + E - E which becomes E - D = D Add D to both sides of the above equation: E - D + D = D + D which makes E = 2×D
Hint #2
In eq.3, replace E with 2×D: F = D + 2×D which makes F = 3×D
Hint #3
In eq.4, substitute 3×D for F: A = D + 3×D which makes A = 4×D
Hint #4
Substitute 4×D for A in eq.6: B + D = 4×D - B In the above equation, subtract D from both sides, and add B to both sides: B + D - D + B = 4×D - B - D + B which makes 2×B = 3×D Divide both sides by 2: 2×B ÷ 2 = 3×D ÷ 2 which makes B = 1½×D
Hint #5
Substitute 1½×D for B, and 3×D for F in eq.2: C = 1½×D + 3×D which makes C = 4½×D
Solution
Substitute 4×D for A, 1½×D for B, 4½×D for C, 2×D for E, and 3×D for F in eq.1: 4×D + 1½×D + 4½×D + D + 2×D + 3×D = 32 which simplifies to 16×D = 32 Divide both sides of the above equation by 16: 16×D ÷ 16 = 32 ÷ 16 which means D = 2 making A = 4×D = 4 × 2 = 8 B = 1½×D = 1½ × 2 = 3 C = 4½×D = 4½ × 2 = 9 E = 2×D = 2 × 2 = 4 F = 3×D = 3 × 2 = 6 and ABCDEF = 839246