Puzzle for November 1, 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 positive integer.
* AB and CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
eq.4 may be written as: 10×A + B = C + D + F which is the same as eq.4a) 10×A + B = C + F + D
Hint #2
In eq.4a, replace C + F with B + D (from eq.2): 10×A + B = B + D + D which becomes 10×A + B = B + 2×D Subtract B from each side of the equation above: 10×A + B - B = B + 2×D - B which makes 10×A = 2×D Divide both sides by 2: 10×A ÷ 2 = 2×D ÷ 2 which makes 5×A = D
Hint #3
In eq.3, replace D with 5×A: E - A = A + 5×A which becomes E - A = 6×A Add A to both sides of the above equation: E - A + A = 6×A + A which makes E = 7×A
Hint #4
In eq.1, substitute 7×A for E: F = A + 7×A which makes F = 8×A
Hint #5
Substitute 5×A for D, 7×A for E, and 8×A for F in eq.6: (5×A + 7×A) ÷ (A + 5×A) = (B + 8×A) ÷ (A + 7×A) which becomes 12×A ÷ 6×A = (B + 8×A) ÷ 8×A which becomes 2 = (B + 8×A) ÷ 8×A Multiply both sides of the above equation by 8×A: 8×A × 2 = 8×A × (B + 8×A) ÷ 8×A which becomes 16×A = B + 8×A Subtract 8×A from each side: 16×A - 8×A = B + 8×A - 8×A which makes 8×A = B
Hint #6
Substitute 8×A for B and F, and 5×A for D in eq.2: 8×A + 5×A = C + 8×A Subtract 8×A from both sides of the above equation: 8×A + 5×A - 8×A = C + 8×A - 8×A which becomes 5×A = C
Hint #7
eq.5 may be written as: eq.5a) A + 10×C + D = E × F
Solution
Substitute (5×A) for C and D, 7×A for E, and 8×A for F in eq.5a: A + 10×(5×A) + (5×A) = 7×A × 8×A which becomes A + 50×A + 5×A = 56×A² which makes 56×A = 56×A² Divide both sides of the above equation by 56×A: 56×A ÷ 56×A = 56×A² ÷ 56×A which means 1 = A making B = F = 8×A = 8 × 1 = 8 C = D = 5×A = 5 × 1 = 5 E = 7×A = 7 × 1 = 7 and ABCDEF = 185578