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| Mind your P and Q (Posted on 2007-02-28) |
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Mr. P's teenaged girlfriend Miss Q is a student of mathematics. Miss Q brought a well framed crochet with a polynomial in X ( having integer coefficients ) stitched on it as the birthday present on his birthday sometime this year. The polynomial is so constructed that one of its roots is equal to Mr. P's age ( in years) on his birthday (this year).
Someone present at the birthday party tried to find its root, not being aware that the root stood for the age of the host. He substituted X=8, which gave him 64. He tried a different number slightly greater than 8 but less than 16, which gave him 70.
If Mr. P is older than his 16- year old girlfriend, how old was he on his birthday this year?
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Submitted by K Sengupta
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Rating: 5.0000 (1 votes)
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Solution:
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Let the polynomial in X be:
f(X) = A0+ A1*X + A2*(X^2)+ .........+An*(x^n)
If Mr. P's age is B and the number tried the second time be L, where L is larger than 8; then, we obtain:
AO+A1*B + A2*(B^2)+..........+ An*(B^n) = 0......(1)
AO+A1*8+ A2*(8^2)+..........+ An*(8^n) = 64-------(2)
AO+A1*L + A2*(L^2)+..........+ An*(L^n) = 70......(3)
clearly, (L-8) is a factor of (70-64) = 6
(B-L) is a factor of 70, and:
(B-8) is a factor of 64.
Accordingly, L = 9, 10, 11, 14 and,
B = 10, 12, 16, 24, 40, 72.
Since, P is older than his 16 year old girlfriend, B cannot be 10, 12 or 16. Also, B cannot be 40 or 72, for in that case (B-L) cannot be a factor of 70.
This, B = 24, L = 10 and B-L = 14.
Consequently, the age of Mr. P on his birthday this year is 24 years.
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Comments: (
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Subject |
Author |
Date |
 | a solutoin | Dej Mar | 2007-03-03 06:24:03 |
 | quesiton | hoodat | 2007-03-01 13:36:45 |
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