Puzzle for September 26, 2023 ( )
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.4, replace B with D + F (from eq.2): D + F + F = A + D which becomes D + 2×F = A + D Subtract D from each side of the equation above: D + 2×F - D = A + D - D which makes 2×F = A
Hint #2
In eq.3, replace A with 2×F: C = 2×F + B + F which becomes eq.3a) C = 3×F + B
Hint #3
In eq.5, substitute (3×F + B) for C (from eq.3a): F - B = B - (3×F + B) which becomes F - B = B - 3×F - B which becomes F - B = -3×F Add B and 3×F to both sides of the above equation: F - B + B + 3×F = -3×F + B + 3×F which makes 4×F = B
Hint #4
Substitute 4×F for B in eq.3a: C = 3×F + 4×F which makes C = 7×F
Hint #5
Substitute 4×F for B in eq.2: 4×F = D + F Subtract F from each side of the above equation: 4×F - F = D + F - F which makes 3×F = D
Hint #6
Substitute 7×F for C, 4×F for B, and 3×F for D in eq.6: 7×F + E = 4×F + 3×F which becomes 7×F + E = 7×F Subtract 7×F from both sides of the equation above: 7×F + E - 7×F = 7×F - 7×F which makes E = 0
Solution
Substitute 2×F for A, 4×F for B, 7×F for C, 3×F for D, and 0 for E in eq.1: 2×F + 4×F + 7×F + 3×F + 0 + F = 17 which simplifies to 17×F = 17 Divide both sides of the above equation by 17: 17×F ÷ 17 = 17 ÷ 17 which means F = 1 making A = 2×F = 2 × 1 = 2 B = 4×F = 4 × 1 = 4 C = 7×F = 7 × 1 = 7 D = 3×F = 3 × 1 = 3 and ABCDEF = 247301