Puzzle for March 7, 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.6, replace C with A + E (from eq.3): A + A + E = E + F which becomes 2×A + E = E + F Subtract E from each side of the equation above: 2×A + E – E = E + F – E which makes 2×A = F
Hint #2
In eq.5, replace F with 2×A: E = A + 2×A which makes E = 3×A
Hint #3
In eq.3, substitute 3×A for E: C = A + 3×A which makes C = 4×A
Hint #4
Substitute 4×A for C in eq.2: B = A + 4×A which makes B = 5×A
Hint #5
Substitute 5×A for B in eq.4: D = A + 5×A which makes D = 6×A
Solution
Substitute 5×A for B, 4×A for C, 6×A for D, 3×A for E, and 2×A for F in eq.1: A + 5×A + 4×A + 6×A + 3×A + 2×A = 21 which simplifies to 21×A = 21 Divide both sides of the above equation by 21: 21×A ÷ 21 = 21 ÷ 21 which means A = 1 making B = 5×A = 5 × 1 = 5 C = 4×A = 4 × 1 = 4 D = 6×A = 6 × 1 = 6 E = 3×A = 3 × 1 = 3 F = 2×A = 2 × 1 = 2 and ABCDEF = 154632