Puzzle for May 31, 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
In eq.5, replace E with D + F (from eq.2): D + F + F = A + D which becomes D + 2×F = A + D Subtract D from both sides of the above equation: D + 2×F – D = A + D – D which makes 2×F = A
Hint #2
In eq.6, replace D with B + C (from eq.3), and A with 2×F: B + C = 2×F + C + F which becomes B + C = 3×F + C Subtract C from each side of the above equation: B + C – C = 3×F + C – C which makes B = 3×F
Hint #3
In eq.4, substitute 2×F for A, and 3×F for B: C = 2×F + 3×F which makes C = 5×F
Hint #4
Substitute 3×F for B, and 5×F for C in eq.3: 3×F + 5×F = D which makes 8×F = D
Hint #5
Substitute 8×F for D in eq.2: E = 8×F + F which makes E = 9×F
Solution
Substitute 2×F for A, 3×F for B, 5×F for C, 8×F for D, and 9×F for E in eq.1: 2×F + 3×F + 5×F + 8×F + 9×F + F = 28 which simplifies to 28×F = 28 Divide both sides of the above equation by 28: 28×F ÷ 28 = 28 ÷ 28 which means F = 1 making A = 2×F = 2 × 1 = 2 B = 3×F = 3 × 1 = 3 C = 5×F = 5 × 1 = 5 D = 8×F = 8 × 1 = 8 E = 9×F = 9 × 1 = 9 and ABCDEF = 235891