Puzzle for April 11, 2022 ( )
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 C + E (from eq.3): C + F = C + E + E which becomes C + F = C + 2×E Subtract C from each side of the equation above: C + F – C = C + 2×E – C which makes F = 2×E
Hint #2
Add F to both sides of eq.6: E + F + F = C – F + F which becomes E + 2×F = C In the above equation, replace F with (2×E): E + 2×(2×E) = C which becomes E + 4×E = C which makes 5×E = C
Hint #3
In eq.3, substitute 5×E for C: A = 5×E + E which makes A = 6×E
Hint #4
Substitute 6×E for A into eq.2: D = 6×E + E which makes D = 7×E
Hint #5
Substitute 7×E for D, and 2×E for F in eq.4: B = 7×E + 2×E which makes B = 9×E
Solution
Substitute 6×E for A, 9×E for B, 5×E for C, 7×E for D, and 2×E for F in eq.1: 6×E + 9×E + 5×E + 7×E + E + 2×E = 30 which simplifies to 30×E = 30 Divide both sides of the above equation by 30: 30×E ÷ 30 = 30 ÷ 30 which means E = 1 making A = 6×E = 6 × 1 = 6 B = 9×E = 9 × 1 = 9 C = 5×E = 5 × 1 = 5 D = 7×E = 7 × 1 = 7 F = 2×E = 2 × 1 = 2 and ABCDEF = 695712