A given year p (expressed in YYYY format) is defined as "Ambiguous" if there exists at least one positive integral solution of the equation x^2 + x + y^2 + 3y = p. Otherwise, the said year (p) is a "Definite" year.
For example, 1890 A.D. was an "Ambiguous" year, since (x,y)=(34,25) corresponds to a positive integral solution of x^2 + x + y^2 + 3y = 1890.
Determine, whether 2006 A.D. is an Ambiguous Year or a Definite Year.
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Submitted by K Sengupta
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Rating: 3.0000 (1 votes)
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Solution:
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2006 A.D. is a definite year.
EXPLANATION:
If possible, let there exist at least one positive integral solution of the equation
x^2 + x + y^2 + 3y = 2006 ............(i)
Multiplying the given equation by 4 and adding 10, we obtain:
A^2 + B^2 = 8034, where A=(2x+1) and B=(2y+3)
Now, square of any integer when divided by 9 yields a remainder equivalent to any one of 0,1,4 and 7 and, accordingly, sum of squares of two positive integers must necessarily possess a remainder equivalent to any one of 0,1,2,4,5,7 and 8 when divided by 9.Consequently, integers possessing either 3 or 6 as remainder upon division by 9 are not expressible as sum of squares of two positive integers.
Now, 8034 possesses a remainder of 6 upon division by 9 so that no inegral solution is possible for Equation(i). This is a contradiction.
Accordingly, 2006 A.D. cannot be an Ambiguous year and consequently, 2006 A.D. must correspond to a Definite year.
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