Puzzle for November 27, 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.5, replace A with E + F (from eq.2): C + F = E + F + E which becomes C + F = 2×E + F Subtract F from each side of the equation above: C + F - F = 2×E + F - F which becomes C = 2×E
Hint #2
In eq.6, replace A with E + F (from eq.2), and C with 2×E: E + F + 2×E = B + F which becomes 3×E + F = B + F Subtract F from each side of the above equation: 3×E + F - F = B + F - F which makes 3×E = B
Hint #3
In eq.3, substitute 3×E for B: 3×E = D + E Subtract E from both sides of the equation above: 3×E - E = D + E - E which makes 2×E = D
Hint #4
Substitute 3×E for B in eq.4: F = 3×E + E which makes F = 4×E
Hint #5
Substitute 4×E for F in eq.2: A = E + 4×E which makes A = 5×E
Solution
Substitute 5×E for A, 3×E for B, 2×E for C and D, and 4×E for F in eq.1: 5×E + 3×E + 2×E + 2×E + E + 4×E = 17 which simplifies to 17×E = 17 Divide both sides of the above equation by 17: 17×E ÷ 17 = 17 ÷ 17 which means E = 1 making A = 5×E = 5 × 1 = 5 B = 3×E = 3 × 1 = 3 C = D = 2×E = 2 × 1 = 2 F = 4×E = 4 × 1 = 4 and ABCDEF = 532214