Puzzle for May 17, 2021 ( )
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 D with A + E (from eq.3): A + B = A + E + E which becomes A + B = A + 2×E Subtract A from each side of the equation above: A + B – A = A + 2×E – A which makes B = 2×E
Hint #2
In eq.4, replace C with B + F (from eq.2): B + B + F = A + F which becomes 2×B + F = A + F Subtract F from each side of the equation above: 2×B + F – F = A + F – F which becomes eq.4a) 2×B = A
Hint #3
In eq.4a, substitute (2×E) for B: 2×(2×E) = A which makes 4×E = A
Hint #4
Substitute 4×E for A in eq.3: D = 4×E + E which makes D = 5×E
Hint #5
In eq.6, substitute 5×E for D: 5×E + E = C – E which becomes 6×E = C – E Add E to both sides of the equation above: 6×E + E = C – E + E which becomes 7×E = C
Hint #6
Substitute 7×E for C, and 2×E for B in eq.2: 7×E = 2×E + F Subtract 2×E from each side of the above equation: 7×E – 2×E = 2×E + F – 2×E which makes 5×E = F
Solution
Substitute 4×E for A, 2×E for B, 7×E for C, and 5×E for D and F in eq.1: 4×E + 2×E + 7×E + 5×E + E + 5×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 4×E = 4 × 1 = 4 B = 2×E = 2 × 1 = 2 C = 7×E = 7 × 1 = 7 D = F = 5×E = 5 × 1 = 5 and ABCDEF = 427515