Puzzle for June 21, 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.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
In eq.4, replace C with E + F (from eq.2): B + F = E + F + E which becomes B + F = 2×E + F Subtract F from each side of the equation above: B + F - F = 2×E + F - F which makes B = 2×E
Hint #2
In eq.5, replace B with 2×E: E - D = 2×E - E which becomes E - D = E Subtract E from each side of the above equation: E - D - E = E - E which makes -D = 0 which means D = 0
Hint #3
In eq.3, substitute 0 for D: F = A - 0 + E which becomes eq.3a) F = A + E
Hint #4
Substitute A + E for F (from eq.3a) in eq.2: C = E + A + E which becomes eq.2a) C = 2×E + A
Hint #5
eq.6 may be written as: 10×E + F - B = A + C In the above equation, substitute A + E for F (from eq.3a), 2×E for B, and 2×E + A for C (from eq.2a): 10×E + A + E - 2×E = A + 2×E + A which becomes 9×E + A = 2×A + 2×E Subtract A and 2×E from both sides: 9×E + A - A - 2×E = 2×A + 2×E - A - 2×E which makes 7×E = A
Hint #6
Substitute 7×E for A in eq.2a: C = 2×E + 7×E which makes C = 9×E
Hint #7
Substitute 7×E for A in eq.3a: F = 7×E + E which makes F = 8×E
Solution
Substitute 7×E for A, 2×E for B, 9×E for C, 0 for D, and 8×E for F in eq.1: 7×E + 2×E + 9×E + 0 + E + 8×E = 27 which simplifies to 27×E = 27 Divide both sides of the above equation by 27: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 7×E = 7 × 1 = 7 B = 2×E = 2 × 1 = 2 C = 9×E = 9 × 1 = 9 F = 8×E = 8 × 1 = 8 and ABCDEF = 729018