Puzzle for October 23, 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.6, replace E with B + C (from eq.3): B - C = B + C - B which becomes B - C = C Add C to both sides of the above equation: B - C + C = C + C which makes B = 2×C
Hint #2
In eq.3, replace B with 2×C: E = 2×C + C which makes E = 3×C
Hint #3
In eq.4, substitute 2×C for B, and 3×C for E: D = 2×C + 3×C which makes D = 5×C
Hint #4
Substitute 5×C for D, and 3×C for E in eq.2: F = 5×C + 3×C which makes F = 8×C
Hint #5
Substitute 8×C for F, and 2×C for B in eq.5: 8×C = A + 2×C Subtract 2×C from each side of the equation above: 8×C - 2×C = A + 2×C - 2×C which makes 6×C = A
Solution
Substitute 6×C for A, 2×C for B, 5×C for D, 3×C for E, and 8×C for F in eq.1: 6×C + 2×C + C + 5×C + 3×C + 8×C = 25 which simplifies to 25×C = 25 Divide both sides of the above equation by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 6×C = 6 × 1 = 6 B = 2×C = 2 × 1 = 2 D = 5×C = 5 × 1 = 5 E = 3×C = 3 × 1 = 3 F = 8×C = 8 × 1 = 8 and ABCDEF = 621538