Puzzle for December 31, 2018 ( )
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 A with C + E (from eq.2): C + E - C = B - E which becomes E = B - E Add E to both sides of the above equation: E + E = B - E + E which makes 2×E = B
Hint #2
In eq.5, substitute 2×E for B: 2×E + E + F = A - E Add E to both sides of the above equation: 2×E + E + F + E = A - E + E which becomes eq.5a) 4×E + F = A
Hint #3
In eq.6, replace A with 4×E + F (from eq.5a): C = 4×E + F + E - F which makes eq.6) C = 5×E
Hint #4
In eq.2, replace C with 5×E: A = 5×E + E which makes A = 6×E
Hint #5
Substitute 6×E for A in eq.5a: 4×E + F = 6×E Subtract 4×E from both sides of the above equation: 4×E + F - 4×E = 6×E - 4×E which makes F = 2×E
Hint #6
Substitute 2×E for B and F in eq.3: 2×E = D + 2×E Subtract 2×E from both sides of the above equation: 2×E - 2×E = D + 2×E - 2×E which makes 0 = D
Solution
Substitute 6×E for A, 2×E for B and F, 5×E for C, and 0 for D in eq.1: 6×E + 2×E + 5×E + 0 + E + 2×E = 16 which simplifies to 16×E = 16 Divide both sides by 16: 16×E ÷ 16 = 16 ÷ 16 which makes E = 1 making A = 6×E = 6 × 1 = 6 B = F = 2×E = 2 × 1 = 2 C = 5×E = 5 × 1 = 5 and ABCDEF = 625012