Puzzle for November 18, 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 + C with A – B (from eq.4): A – B + F = A – F In the above equation, subtract A from both sides, and add B and F to both sides: A – B + F – A + B + F = A – F – A + B + F which simplifies to 2×F = B
Hint #2
In eq.6, replace B with 2×F: C – F = 2×F + F which becomes C – F = 3×F Add F to both sides of the above equation: C – F + F = 3×F + F which makes C = 4×F
Hint #3
In eq.4, substitute 2×F for B, and 4×F for C: A – 2×F = 2×F + 4×F which becomes A – 2×F = 6×F Add 2×F to both sides of the above equation: A – 2×F + 2×F = 6×F + 2×F which makes A = 8×F
Hint #4
Substitute 2×F for B, 8×F for A, and 4×F for C in eq.5: 2×F + D – E = 8×F – 2×F – 4×F which becomes 2×F + D – E = 2×F In the above equation, add E to each side, and subtract 2×F from both sides: 2×F + D – E + E – 2×F = 2×F + E – 2×F which simplifies to D = E
Hint #5
Substitute D for E, 4×F for C, and 8×F for A in eq.3: D + D – 4×F = 8×F + 4×F which becomes 2×D – 4×F = 12×F Add 4×F to each side of the equation above: 2×D – 4×F + 4×F = 12×F + 4×F which makes 2×D = 16×F Divide both sides by 2: 2×D ÷ 2 = 16×F ÷ 2 which means D = 8×F which also means E = D = 8×F
Solution
Substitute 8×F for A and D and E, 2×F for B, and 4×F for C in eq.1: 8×F + 2×F + 4×F + 8×F + 8×F + F = 31 which simplifies to 31×F = 31 Divide both sides of the equation above by 31: 31×F ÷ 31 = 31 ÷ 31 which means F = 1 making A = D = E = 8×F = 8×1 = 8 B = 2×F = 2×1 = 2 C = 4×F = 4×1 = 4 and ABCDEF = 824881