Puzzle for August 25, 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
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Hint #1
Add C to both sides of eq.5: C + E + C = B – C + C which becomes 2×C + E = B In the above equation, replace B with A + E (from eq.2): 2×C + E = A + E Subtract E from each side: 2×C + E – E = A + E – E which makes 2×C = A
Hint #2
In eq.4, replace A with 2×C: E – F = 2×C – C which becomes eq.4a) E – F = C
Hint #3
In eq.3, substitute E – F for C (from eq.4a): D + E = E – F + F which becomes D + E = E Subtract E from each side of the above equation: D + E – E = E – E which makes D = 0
Hint #4
Substitute 0 for D, and A + E for B (from eq.2) in eq.6: 0 + E + F = A + A + E + C – 0 which becomes E + F = 2×A + E + C Subtract E from both sides of the equation above: E + F – E = 2×A + E + C – E which becomes eq.6a) F = 2×A + C
Hint #5
Substitute (2×C) for A in eq.6a: F = 2×(2×C) + C which becomes F = 4×C + C which makes F = 5×C
Hint #6
Substitute 5×C for F in eq.4a: E – 5×C = C Add 5×C to both sides of the above equation: E – 5×C + 5×C = C + 5×C which makes E = 6×C
Hint #7
Substitute 2×C for A, and 6×C for E in eq.2: B = 2×C + 6×C which makes B = 8×C
Solution
Substitute 2×C for A, 8×C for B, 0 for D, 6×C for E, and 5×C for F in eq.1: 2×C + 8×C + C + 0 + 6×C + 5×C = 22 which simplifies to 22×C = 22 Divide both sides of the above equation by 22: 22×C ÷ 22 = 22 ÷ 22 which means C = 1 making A = 2×C = 2 × 1 = 2 B = 8×C = 8 × 1 = 8 E = 6×C = 6 × 1 = 6 F = 5×C = 5 × 1 = 5 and ABCDEF = 281065