Puzzle for May 2, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, B, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add A + D to both sides of eq.6: B - A + A + D = E - D + A + D which becomes B + D = E + A In eq.2, replace B + D with E + A: A + E + F = E + A Subtract both A and E from each side of the above equation: A + E + F - A - E = E + A - A - E which simplifies to F = 0
Hint #2
In eq.6, replace B - A with C - E (from eq.3): C - E = E - D Add both D and E to each side of the equation above: C - E + D + E = E - D + D + E which simplifies to eq.6a) C + D = 2×E
Hint #3
Add C to both sides of eq.5: D + E + C = B - C + C which can be written as C + D + E = B In the equation above, substitute 2×E for C + D (from eq.6a): 2×E + E = B which makes 3×E = B
Hint #4
Add C + A to both sides of eq.4: E - A + C + A = D - C + C + A which becomes eq.4a) E + C = D + A Add the left and right sides of eq.4a to the left and right sides of eq.3, respectively: C - E + E + C = B - A + D + A which becomes eq.3a) 2×C = B + D
Hint #5
Subtract C from each side of eq.6a: C + D - C = 2×E - C which becomes eq.6b) D = 2×E - C Substitute 3×E for B, and 2×E - C for D (from eq.6b) in eq.3a: 2×C = 3×E + 2×E - C Add C to both sides of the above equation: 2×C + C = 3×E + 2×E - C + C which simplifies to 3×C = 5×E Divide both sides by 3: 3×C ÷ 3 = 5×E ÷ 3 which makes C = 1⅔×E
Hint #6
Substitute 1⅔×E for C in eq.6b: D = 2×E - 1⅔×E which makes D = ⅓×E
Hint #7
Substitute 1⅔×E for C, and ⅓×E for D in eq.4a: E + 1⅔×E = ⅓×E + A Subtract ⅓×E from both sides of the above equation: E + 1⅔×E - ⅓×E = ⅓×E + A - ⅓×E which makes 2⅓×E = A
Solution
Substitute 2⅓×E for A, 3×E for B, 1⅔×E for C, ⅓×E for D, and 0 for F in eq.1: 2⅓×E + 3×E + 1⅔×E + ⅓×E + E + 0 = 25 which simplifies to 8⅓×E = 25 Divide each side by 8⅓: 8⅓×E ÷ 8⅓ = 25 ÷ 8⅓ which makes E = 3 making A = 2⅓×E = 2⅓ × 3 = 7 B = 3×E = 3 × 3 = 9 C = 1⅔×E = 1⅔ × 3 = 5 D = ⅓×E = ⅓ × 3 = 1 and ABCDEF = 795130