Puzzle for April 19, 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.3, replace C with A + B (from eq.2): E = A + A + B which becomes eq.3a) E = 2×A + B
Hint #2
In eq.5, replace E with 2×A + B (from eq.3a): B = 2×A + B – F In the above equation, subtract B from both sides, and add F to both sides: B – B + F = 2×A + B – F – B + F which makes F = 2×A
Hint #3
In eq.4, substitute 2×A for F: D = A + 2×A which makes D = 3×A
Hint #4
Substitute 3×A for D, and 2×A for F in eq.6: A = B – 3×A – 2×A which becomes A = B – 5×A Add 5×A to both sides of the above equation: A + 5×A = B – 5×A + 5×A which makes 6×A = B
Hint #5
Substitute 6×A for B, and 2×A for F in eq.5: 6×A = E – 2×A Add 2×A to both sides of the equation above: 6×A + 2×A = E – 2×A + 2×A which makes 8×A = E
Hint #6
Substitute 6×A for B in eq.2: C = A + 6×A which makes C = 7×A
Solution
Substitute 6×A for B, 7×A for C, 3×A for D, 8×A for E, and 2×A for F in eq.1: A + 6×A + 7×A + 3×A + 8×A + 2×A = 27 which simplifies to 27×A = 27 Divide both sides of the above equation by 27: 27×A ÷ 27 = 27 ÷ 27 which means A = 1 making B = 6×A = 6 × 1 = 6 C = 7×A = 7 × 1 = 7 D = 3×A = 3 × 1 = 3 E = 8×A = 8 × 1 = 8 F = 2×A = 2 × 1 = 2 and ABCDEF = 167382