Puzzle for April 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.3, replace D with C + E (from eq.2): F = C + E + E which becomes eq.3a) F = C + 2×E
Hint #2
In eq.6, replace F with C + 2×E (from eq.3a): E + C + 2×E = B + C which becomes 3×E + C = B + C Subtract C from each side of the equation above: 3×E + C - C = B + C - C which makes 3×E = B
Hint #3
In eq.4, substitute 3×E for B: A = 3×E + E which makes A = 4×E
Hint #4
Substitute 4×E for A in eq.5: C - 4×E = 4×E - E which becomes C - 4×E = 3×E Add 4×E to both sides of the above equation: C - 4×E + 4×E = 3×E + 4×E which makes C = 7×E
Hint #5
Substitute 7×E for C in eq.2: D = 7×E + E which makes D = 8×E
Hint #6
Substitute 3×E for B, and 7×E for C in eq.6: E + F = 3×E + 7×E which becomes E + F = 10×E Subtract E from each side of the above equation: E + F - E = 10×E - E which makes F = 9×E
Solution
Substitute 4×E for A, 3×E for B, 7×E for C, 8×E for D, and 9×E for F in eq.1: 4×E + 3×E + 7×E + 8×E + E + 9×E = 32 which simplifies to 32×E = 32 Divide both sides of the above equation by 32: 32×E ÷ 32 = 32 ÷ 32 which means E = 1 making A = 4×E = 4 × 1 = 4 B = 3×E = 3 × 1 = 3 C = 7×E = 7 × 1 = 7 D = 8×E = 8 × 1 = 8 F = 9×E = 9 × 1 = 9 and ABCDEF = 437819