.If a and b are positive integers such that a2 - b4 = 2009, find a + b

1.	.If a and b are positive integers such that a2 - b4 = 2009, find a + b.
Answer» a² - b⁴ = (a + b²)(a - b²) Now find factors of 2009.Factors of 2009 are = 1, 7, 41,49, 287,2009 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 'u' and 'v' such that uv = 2009, a + b² = u and a - b² = v, then 2b² = u-v. If u = 2009, then v = 1 and 2b² = 2008, then, b² = 2008/2 b = √1004 which is not and integer. Now, if u = 287, then v= 7 and 2b² = 280 then, b² = 280/2 b = √140 which is also not an integer. Now, if u = 49, then v= 41 and 2b² = 8 then b² = 8/2 b = √4 b = 2 ……(1) Now substitute value of b =2 in a² - b⁴ = 2009 a² - 2⁴ = 2009 a² - 16 = 2009 a² = 2009 + 16 a² = 2025 a = √2025 a = 45…..(2) So a+b=45+2=47.

.If a and b are positive integers such that a2 - b4 = 2009, find a + b.

Answer»

a² - b⁴ = (a + b²)(a - b²)

Now find factors of 2009.Factors of 2009 are = 1, 7, 41,49, 287,2009

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 'u' and 'v' such that uv = 2009, a + b² = u and a - b² = v, then 2b² = u-v.

If u = 2009, then v = 1 and 2b² = 2008,

then, b² = 2008/2

b = √1004 which is not and integer.

Now, if u = 287, then v= 7 and 2b² = 280

then, b² = 280/2

b = √140 which is also not an integer.

Now, if u = 49, then v= 41 and 2b² = 8

then b² = 8/2

b = √4

b = 2 ……(1)

Now substitute value of b =2 in a² - b⁴ = 2009

a² - 2⁴ = 2009

a² - 16 = 2009

a² = 2009 + 16

a² = 2025

a = √2025

a = 45…..(2)

So a+b=45+2=47.