Puzzle for July 4, 2019 ( )
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
Add A to both sides of eq.6: D - A + A = A + E + A which becomes eq.6a) D = 2×A + E In eq.4, replace D with 2×A + E (from eq.6a): A + F = 2×A + E + E which becomes A + F = 2×A + 2×E Subtract A from both sides of the above equation: A + F - A = 2×A + 2×E - A which becomes eq.4a) F = A + 2×E
Hint #2
In eq.2, replace F with A + 2×E (from eq.4a): A + 2×E = A + C Subtract A from both sides of the above equation: A + 2×E - A = A + C - A 2×E = C
Hint #3
In eq.3, substitute 2×E for C, and A + 2×E for F (from eq.4a): 2×E + A + 2×E = D - E which is the same as A + 4×E = D - E Add E to both sides of the equation above: A + 4×E + E = D - E + E which becomes eq.3a) A + 5×E = D
Hint #4
In eq.3a, substitute 2×A + E for D (from eq.6a): A + 5×E = 2×A + E Subtract both A and E from each side of the above equation: A + 5×E - A - E = 2×A + E - A - E which simplifies to 4×E = A
Hint #5
Substitute (4×E) for A in eq.6a: D = 2×(4×E) + E which is equivalent to D = 8×E + E which makes D = 9×E
Hint #6
Substitute 4×E for A in eq.4a: F = 4×E + 2×E which makes F = 6×E
Hint #7
Substitute 2×E for C, and 4×E for A in eq.5: B - 2×E = 4×E + 2×E + E which becomes B - 2×E = 7×E Add 2×E to both sides of the above equation: B - 2×E + 2×E = 7×E + 2×E which makes B = 9×E
Solution
Substitute 4×E for A, 9×E for B and D, 2×E for C, and 6×E for F in eq.1: 4×E + 9×E + 2×E + 9×E + E + 6×E = 31 which simplifies to 31×E = 31 Divide both sides of the above equation by 31: 31×E ÷ 31 = 31 ÷ 31 which means E = 1 making A = 4×E = 4 × 1 = 4 B = D = 9×E = 9 × 1 = 9 C = 2×E = 2 × 1 = 2 F = 6×E = 6 × 1 = 6 and ABCDEF = 492916