QJEEM20Ma0554: JEE Main 2020 Mathematics - Class 11 JEEM20Ma0554 Solution: The greatest positive integer k, for which 49 k + 1 is a factor …

Question

Verified Solution • 2025–26 Session moderate

The greatest positive integer k, for which 49^k + 1 is a factor of the sum
49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

Accepted Answer

1 + 49 + 49² + ..... + 49¹²⁵

sum of G.P. =

\frac{1. (49^{126} - 1)}{49 - 1}

=

\frac{(49^{63} + 1) (49^{63} - 1)}{48}

Also 49⁶³ - 1

= (1 + 48)⁶³ - 1

= [⁶³C₀

\times

1 + ⁶³C₁

\times

48 + ⁶³C₂

\times

(48)² + .... ] - 1

= [1 + 48

λ

] - 1 = 48

λ

So

\frac{(49^{63} - 1)}{48}

= integer

∴

49⁶³ + 1 is a factor.

So k = 63.

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