Puzzle for April 19, 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.2, replace B with C + E (from eq.3): D = C + E + E which becomes eq.2a) D = C + 2×E
Hint #2
In eq.5, replace D with C + 2×E (from eq.2a): C + F = C + 2×E + E which becomes C + F = C + 3×E Subtract C from each side of the equation above: C + F - C = C + 3×E - C which makes F = 3×E
Hint #3
In eq.4, substitute 3×E for F: 3×E - E = D - A which becomes 2×E = D - A Add A to both sides of the above equation: 2×E + A = D - A + A which becomes eq.4a) 2×E + A = D
Hint #4
Substitute C + 2×E for D (from eq.2a) into eq.4a: 2×E + A = C + 2×E Subtract 2×E from each side of the equation above: 2×E + A - 2×E = C + 2×E - 2×E which makes A = C
Hint #5
Substitute C + 2×E for D (from eq.2a), and C + E for B (from eq.3) in eq.6: C + 2×E + E = C + C + E - E which becomes C + 3×E = 2×C Subtract C from both sides of the equation above: C + 3×E - C = 2×C - C which makes 3×E = C and also makes 3×E = C = A
Hint #6
Substitute 3×E for C in eq.2a: D = 3×E + 2×E which makes D = 5×E
Hint #7
Substitute 3×E for C in eq.3: B = 3×E + E which makes B = 4×E
Solution
Substitute 3×E for A and C and F, 4×E for B, and 5×E for D in eq.1: 3×E + 4×E + 3×E + 5×E + E + 3×E = 19 which simplifies to 19×E = 19 Divide both sides of the above equation by 19: 19×E ÷ 19 = 19 ÷ 19 which means E = 1 making A = C = F = 3×E = 3 × 1 = 3 B = 4×E = 4 × 1 = 4 D = 5×E = 5 × 1 = 5 and ABCDEF = 343513