Puzzle for April 17, 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.
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Hint #1
In eq.5, replace B with D + F (from eq.2): D - F = D + F - D which becomes D - F = F Add F to both sides of the above equation: D - F + F = F + F which makes D = 2×F
Hint #2
In eq.2, replace D with 2×F: B = 2×F + F which makes B = 3×F
Hint #3
In eq.6, substitute A + B for E (from eq.3): A - B = A + B - A which becomes A - B = B Add B to both sides of the equation above: A - B + B = B + B which makes eq.6a) A = 2×B
Hint #4
Substitute 3×F for B in eq.6a: A = 2×(3×F) which makes A = 6×F
Hint #5
Substitute 6×F for A in eq.4: C = 6×F + F which makes C = 7×F
Hint #6
Substitute 6×F for A, and 3×F for B in eq.3: E = 6×F + 3×F which makes E = 9×F
Solution
Substitute 6×F for A, 3×F for B, 7×F for C, 2×F for D, and 9×F for E in eq.1: 6×F + 3×F + 7×F + 2×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 = 6×F = 6 × 1 = 6 B = 3×F = 3 × 1 = 3 C = 7×F = 7 × 1 = 7 D = 2×F = 2 × 1 = 2 E = 9×F = 9 × 1 = 9 and ABCDEF = 637291