Consider the equation \((146)_π+(313)_{πβ2}=(246)_8\). Which of theβ¦
2019
Consider the equationΒ \((146)_π+(313)_{πβ2}=(246)_8\). Which of the following is the value ofΒ \(π\)?
- A.
8
- B.
7
- C.
10
- D.
16
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Correct answer: B
Compute the right-hand side: (246)_8 = 2Β·8^2 + 4Β·8 + 6 = 166.
Express the left side in terms of b: (146)_b = b^2 + 4b + 6.
Express (313)_{b-2}: 3(b-2)^2 + (b-2) + 3 = 3b^2 - 11b + 13.
Add them and set equal to 166: (b^2 + 4b + 6) + (3b^2 - 11b + 13) = 166, which simplifies to 4b^2 - 7b + 19 = 166.
Solve the quadratic: 4b^2 - 7b - 147 = 0. The discriminant is 49 + 2352 = 2401 = 49^2, so b = (7 Β± 49)/8.
The positive solution is b = (7 + 49)/8 = 7 (the other root is negative and not a valid base).
Check base validity: digits in (146)_b require b > 6 and digits in (313)_{b-2} require b-2 > 3 (so b > 5). Thus b = 7 is valid.
Answer: 7
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