Puzzle for September 25, 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 D + F (from eq.3): D + E = D + F - E In the above equation, subtract D from both sides, and add E to both sides: D + E - D + E = D + F - E - D + E which makes 2×E = F
Hint #2
In eq.6, replace F with 2×E: E + 2×E = B - E which becomes 3×E = B - E Add E to both sides of the above equation: 3×E + E = B - E + E which makes 4×E = B
Hint #3
In eq.2, substitute 4×E for B, and 2×E for F: 4×E = C + 2×E Subtract 2×E from each side of the equation above: 4×E - 2×E = C + 2×E - 2×E which makes 2×E = C
Hint #4
Substitute 4×E for B, and 2×E for C in eq.4: D = 4×E + 2×E + E which makes D = 7×E
Hint #5
Substitute 7×E for D, and 2×E for F in eq.3: A = 7×E + 2×E which makes A = 9×E
Solution
Substitute 9×E for A, 4×E for B, 2×E for C and F, and 7×E for D in eq.1: 9×E + 4×E + 2×E + 7×E + E + 2×E = 25 which simplifies to 25×E = 25 Divide both sides of the above equation by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 4×E = 4 × 1 = 4 C = F = 2×E = 2 × 1 = 2 D = 7×E = 7 × 1 = 7 and ABCDEF = 942712