The number $(101)^{100} - 1$ is divisible by
- $10^4$
- $10^6$
- $10^8$
- $10^{12}$
The Correct Option is A
Solution and Explanation
$={ }^{100} C_{1} 100+{ }^{100} C_{2} 100^{2}+\ldots+{ }^{100} C_{100} 100^{100}$
$=(100)^{2}\left[1+{ }^{100} C_{2}+{ }^{100} C_{3} 1000+\ldots\right]$
Hence, it is divisible by $10^{4}$.
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Concepts Used:
Binomial Expansion Formula
The binomial expansion formula involves binomial coefficients which are of the form
(n/k)(or) nCk and it is calculated using the formula, nCk =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem. Here are the binomial expansion formulas.
This binomial expansion formula gives the expansion of (x + y)n where 'n' is a natural number. The expansion of (x + y)n has (n + 1) terms. This formula says:
We have (x + y)n = nC0 xn + nC1 xn-1 . y + nC2 xn-2 . y2 + … + nCn yn
General Term = Tr+1 = nCr xn-r . yr
- General Term in (1 + x)n is nCr xr
- In the binomial expansion of (x + y)n , the rth term from end is (n – r + 2)th .