Puzzle for November 21, 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 E with D + F (from eq.5): C + D = D + F + F which becomes C + D = D + 2×F Subtract D from each side of the equation above: C + D - D = D + 2×F - D which makes C = 2×F
Hint #2
In eq.4, replace C with 2×F: A = 2×F + F which makes A = 3×F
Hint #3
In eq.3, substitute 3×F for A, and 2×F for C: E = 3×F + 2×F which makes E = 5×F
Hint #4
Substitute 2×F for C, and 5×F for E in eq.2: B = 2×F + 5×F which makes B = 7×F
Hint #5
Substitute 5×F for E in eq.5: 5×F = D + F Subtract F from each side of the above equation: 5×F - F = D + F - F which makes 4×F = D
Solution
Substitute 3×F for A, 7×F for B, 2×F for C, 4×F for D, and 5×F for E in eq.1: 3×F + 7×F + 2×F + 4×F + 5×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 = 3×F = 3 × 1 = 3 B = 7×F = 7 × 1 = 7 C = 2×F = 2 × 1 = 2 D = 4×F = 4 × 1 = 4 E = 5×F = 5 × 1 = 5 and ABCDEF = 372451