9.2.Io a and b are positive integers such that a - b = 2009, find a +

1.	9.2.Io a and b are positive integers such that a - b = 2009, find a + b.20- 13. is
Answer» a² - b⁴ = 2009 a² - b⁴ = (a + b²)(a - b²)Now, we have to find out all the factors of 2009. Factors of 2009 are = 1, 7, 41,49, 287 and 2009 Now, we have to use the difference of squares of factorization to obtain (a + b²)(a - b²) = 2009 The prime factorization of 2009 is 7²*41. If we choose two factors 'x' and 'y' such that xy = 2009, a + b² = x and a - b² = y, then 2b² = x - y. If x = 2009, then y = 1 and 2b² = 2008, then, b² = 2008/2b =√1004 which is not and integer. Now, if x = 287, then y = 7 and 2b² = 280then, b² = 280/2b =√140 which is also not an integer. Now, if x = 49, then y = 41 and 2b² = 8 then b² = 8/2b =√4b = 2 Now, putting the value of b =2 in a² - b⁴ = 2009 a² - 2⁴ = 2009 a² - 16 = 2009 a² = 2009 + 16 a² = 2025 a =√2025a = 45 So, the value of a and b is 45 and 2 respectively.

9.2.Io a and b are positive integers such that a - b = 2009, find a + b.20- 13. is

Answer»

a² - b⁴ = 2009

a² - b⁴ = (a + b²)(a - b²)Now, we have to find out all the factors of 2009.

Factors of 2009 are = 1, 7, 41,49, 287 and 2009

Now, we have to use the difference of squares of factorization to obtain (a + b²)(a - b²) = 2009

The prime factorization of 2009 is 7²*41.

If we choose two factors 'x' and 'y' such that xy = 2009, a + b² = x and a - b² = y, then 2b² = x - y.

If x = 2009, then y = 1 and 2b² = 2008, then, b² = 2008/2b =√1004 which is not and integer.

Now, if x = 287, then y = 7 and 2b² = 280then,

b² = 280/2b =√140 which is also not an integer.

Now, if x = 49, then y = 41 and 2b² = 8

then b² = 8/2b =√4b = 2

Now, putting the value of b =2 in a² - b⁴ = 2009 a² - 2⁴ = 2009 a² - 16 = 2009 a² = 2009 + 16 a² = 2025 a =√2025a = 45

So, the value of a and b is 45 and 2 respectively.