Puzzle for January 25, 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
In eq.2, replace B with E + F (from eq.3): A + E + F = D In eq.4, replace D with A + E + F: A + A + E + F = C + F Subtract F from each side of the above equation: A + A + E + F - F = C + F - F which becomes eq.4a) 2×A + E = C
Hint #2
In eq.5, substitute 2×A + E for C (from eq.4a): A + 2×A + E = D + E Subtract E from each side of the equation above: A + 2×A + E - E = D + E - E which simplifies to 3×A = D
Hint #3
In eq.2, replace D with 3×A: A + B = 3×A Subtract A from both sides of the equation above: A + B - A = 3×A - A which makes B = 2×A
Hint #4
Add A + F to both sides of eq.6: A + B - F + A + F = C - A + F + A + F which becomes 2×A + B = C + 2×F In the above equation, replace B with 2×A: 2×A + 2×A = C + 2×F which becomes 4×A = C + 2×F which may also be written as eq.6a) 4×A = C + F + F
Hint #5
Substitute A + D for C + F (from eq.4) into eq.6a: 4×A = A + D + F Replace D with 3×A in the equation above: 4×A = A + 3×A + F which becomes 4×A = 4×A + F Subtract 4×A from each side: 4×A - 4×A = 4×A + F - 4×A which means 0 = F
Hint #6
Substitute 3×A for D, and 0 for F in eq.4: A + 3×A = C + 0 which means 4×A = C
Hint #7
Substitute 3×A for D, and 4×A for C in eq.5: A + 4×A = 3×A + E Subtract 3×A from each side of the equation above: A + 4×A - 3×A = 3×A + E - 3×A which makes 2×A = E
Solution
Substitute 2×A for B and E, and 4×A for C, 3×A for D, and 0 for F in eq.1: A + 2×A + 4×A + 3×A + 2×A + 0 = 24 which simplifies to 12×A = 24 Divide both sides of the equation above by 12: 12×A ÷ 12 = 24 ÷ 12 which means A = 2 making B = E = 2×A = 2 × 2 = 4 C = 4×A = 4 × 2 = 8 D = 3×A = 3 × 2 = 6 and ABCDEF = 248640