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  4. The number (492 -4) (492 -49) is divisible by ..................

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Q. The number $(49^2 -4) (49^2 -49)$ is divisible by ..................

KCETKCET 2010Binomial Theorem

Solution:

Given, $\left(49^{2}-4\right)\left(49^{3}-49\right)$
$=\left[(49)^{2}-(2)^{2}\right]\left[(49)^{2}-1\right] \cdot 49$
$=(49+2)(49-2)(49+1)(49-1) \cdot 49$
$=51 \cdot 47 \cdot 50 \cdot 48 \cdot 49$
$=(51 \cdot 50 \cdot 49 \cdot 48 \cdot 47)$
Which is the product of five consecutive integers and hence it divisible by $5 !$.