Question

Find the number of ordered pairs (x, y) of positive
integers such that (x + y)(x2 + 9y) is the cube of a prime
number.​

Answer 1

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Given, x+y=1000.

Then x=10,20,30,_ _ _ _ _,990 and

y=990,980,970,_ _ _ _ _10.

∴ no. of ordered pair (x,y) will be, no. of terms in the series 10+20+30+_ _ _ _+990.

Here, this series is in A.P. with first terms (a)=10 and common difference (d)=10 and nth term (an)=990

Since,

an =a+(n−1)d [Where, n is the number of terms in the given series.]

or, 990=10+(n−1)10

or, 980=10(n−1)

or, n=99

∴ total no. of order pairs (x,y)=99.

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