Let f ( x ) = 2 n + R , n N , a n d f ( 4 ) = 133 , f ( 5 ) = 255 . Then the sum of all the positive integer divisors of ( f ( 3 ) - f ( 2 ) ) is [2023]

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Solution

Q.
Let $f (x) = 2 x^{n} + λ, λ \in R, n \in N, a n d f (4) = 133, f (5) = 255 .$ Then the sum of all the positive integer divisors of $(f (3) - f (2))$ is [2023]

1 59

2 60

3 61

4 58

Ans.
(2)

$f (x) = 2 x^{n} + λ, λ \in R, n \in N$

$f (4) = 2 {(4)}^{n} + λ = 133$ ... $(i)$

$f (5) = 2 {(5)}^{n} + λ = 255$ ... $(ii)$

(ii) - (i), we get

$\Rightarrow f (5) - f (4) = 2 {(5)}^{n} - 2 {(4)}^{n} = 122; 5^{n} - 4^{n} = 61$

which gives $n = 3$ .

Then $f (3) = 2 {(3)}^{3} + λ, f (2) = 2 {(2)}^{3} + λ$

$f (3) = 54 + λ,$ $f (2) = 16 + λ .$ So, $f (3) - f (2) = 38$

Positive divisions of 38 are 1, 2, 19, 38 and whose sum = 1 + 2 + 19 + 38 = 60

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