Puzzle for October 4, 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.
* BC and AB are 2-digit numbers (not B×C or A×B).
Scratchpad
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Hint #1
In eq.2, replace E with C + F (from eq.3): A = C + F - F which makes A = C
Hint #2
In eq.4, replace C with A: A - F = A + F In the above equation, subtract A from both sides, and add F to both sides: A - F - A + F = A + F - A + F which makes 0 = 2×F which means 0 = F
Hint #3
In eq.2, substitute 0 for F : A = E - 0 which makes A = E
Hint #4
Substitute A for C and E, and 0 for F in eq.5: A + B = A + D + A + 0 which becomes A + B = 2×A + D Subtract A and D from both sides of the equation above: A + B - A - D = 2×A + D - A - D which becomes B - D = A and also makes eq.5a) B - D = A = C = E
Hint #5
eq.6 may be written as: 10×B + C - (10×A + B) = B + D which becomes 10×B + C - 10×A - B = B + D which becomes eq.6a) 9×B + C - 10×A = B + D
Hint #6
Substitute (B - D) for C and A (from eq.5a) in eq.6a: 9×B + (B - D) - 10×(B - D) = B + D which becomes 9×B + B - D - 10×B + 10×D = B + D which becomes 9×D = B + D Subtract D from each side of the equation above: 9×D - D = B + D - D which becomes 8×D = B
Hint #7
Substitute 8×D for B in eq.5a: 8×D - D = A = C = E which makes 7×D = A = C = E
Solution
Substitute 7×D for A and C and E, 8×D for B, and 0 for F in eq.1: 7×D + 8×D + 7×D + D + 7×D + 0 = 30 which simplifies to 30×D = 30 Divide both sides of the above equation by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = C = E = 7×D = 7 × 1 = 7 B = 8×D = 8 × 1 = 8 and ABCDEF = 787170