Puzzle for September 26, 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
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Hint #1
In eq.6, replace C with D + F (from eq.4): 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.4, replace D with 2×F: C = 2×F + F which makes C = 3×F
Hint #3
In eq.2, substitute 3×F for C, and 2×F for D: A = 3×F + 2×F which makes A = 5×F
Hint #4
In eq.5, substitute 3×F for C, and 2×F for D: E = 3×F + 2×F + F which makes E = 6×F
Hint #5
Substitute 6×F for E in eq.3: 6×F = B + F Subtract F from each side of the above equation: 6×F - F = B + F - F which makes 5×F = B
Solution
Substitute 5×F for A and B, 3×F for C, 2×F for D, and 6×F for E in eq.1: 5×F + 5×F + 3×F + 2×F + 6×F + F = 22 which simplifies to 22×F = 22 Divide both sides of the above equation by 22: 22×F ÷ 22 = 22 ÷ 22 which means F = 1 making A = B = 5×F = 5 × 1 = 5 C = 3×F = 3 × 1 = 3 D = 2×F = 2 × 1 = 2 E = 6×F = 6 × 1 = 6 and ABCDEF = 553261