Puzzle for March 14, 2022 ( )
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 A with C + D (from eq.4): D + E = C + D + C which becomes D + E = 2×C + D Subtract D from each side of the above equation: D + E – D = 2×C + D – D which makes E = 2×C
Hint #2
In eq.6, replace E with 2×C: C + 2×C = D – 2×C which becomes 3×C = D – 2×C Add 2×C to both sides of the above equation: 3×C + 2×C = D – 2×C + 2×C which makes 5×C = D
Hint #3
In eq.4, substitute 5×C for D: A = C + 5×C which makes A = 6×C
Hint #4
Substitute 5×C for D in eq.2: 5×C = B + C Subtract C from each side of the equation above: 5×C – C = B + C – C which makes 4×C = B
Hint #5
Substitute 4×C for B, and 5×C for D in eq.3: F = 4×C + 5×C which makes F = 9×C
Solution
Substitute 6×C for A, 4×C for B, 5×C for D, 2×C for E, and 9×C for F in eq.1: 6×C + 4×C + C + 5×C + 2×C + 9×C = 27 which simplifies to 27×C = 27 Divide both sides of the above equation by 27: 27×C ÷ 27 = 27 ÷ 27 which means C = 1 making A = 6×C = 6 × 1 = 6 B = 4×C = 4 × 1 = 4 D = 5×C = 5 × 1 = 5 E = 2×C = 2 × 1 = 2 F = 9×C = 9 × 1 = 9 and ABCDEF = 641529