Puzzle for May 24, 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.6 may be written as: A + E = C + B + D In the above equation, replace B + D with C + E (from eq.3): A + E = C + C + E which becomes A + E = 2×C + E Subtract E from both sides of the equation above: A + E – E = 2×C + E – E which makes A = 2×C
Hint #2
In eq.2, replace A with 2×C: B = 2×C + C which makes B = 3×C
Hint #3
In eq.5, substitute 3×C for B: E – C = 3×C + C Add C to both sides of the above equation: E – C + C = 4×C + C which makes E = 5×C
Hint #4
Substitute 5×C for E, and 3×C for B in eq.3: C + 5×C = 3×C + D which becomes 6×C = 3×C + D Subtract 3×C from each side of the equation above: 6×C – 3×C = 3×C + D – 3×C which makes 3×C = D
Hint #5
In eq.4, substitute 3×C for D, 5×C for E, and 2×C for A: 3×C + 5×C = 2×C + F which becomes 8×C = 2×C + F Subtract 2×C from both sides of the above equation: 8×C – 2×C = 2×C + F – 2×C which makes 6×C = F
Solution
Substitute 2×C for A, 3×C for B and D, 5×C for E, and 6×C for F in eq.1: 2×C + 3×C + C + 3×C + 5×C + 6×C = 20 which simplifies to 20×C = 20 Divide both sides of the above equation by 20: 20×C ÷ 20 = 20 ÷ 20 which means C = 1 making A = 2×C = 2 × 1 = 2 B = D = 3×C = 3 × 1 = 3 E = 5×C = 5 × 1 = 5 F = 6×C = 6 × 1 = 6 and ABCDEF = 231356