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Q. If $n \in N$, then $121^n - 25^n + 1900^n -(-4)^n$ is divisible by

Binomial Theorem

Solution:

$121^n - 25^n = (96 + 25)^n - 25^n$ is divisible by $96$.
$1900^n - (- 4)^n = (1904 - 4)^n - (- 4)^n$ is divisible by $1904$. Hence, both the above are divisible by $16$.
Further, $121^n - (- 4)^n$ is divisible by $125$ and $1900^n - 25^n$ is divisible by $1875$. Hence, both the above are divisible by $125$.
$\therefore $ The given number is divisible by $16 \times 125 = 2000$.