If $a, b, p \in {N}$ and $p$ is prime, then $(a+b)^p-a^p-b^p$ is divisible by

JEE MATHEMATICS MCQ: BINOMIAL THEOREM

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Application +4 | -1

a, b, p \in {N}

and

p

is prime, then

(a+b)^p-a^p-b^p

is divisible by

View answer and explanation

Correct option: A

p

Option A (

p

) is correct because

(a+b)^p-a^p-b^p

can be expressed as a sum of terms, each of which is divisible by

p

due to the properties of binomial coefficients and prime numbers.

Practice All Chapter MCQs →

\left({ }^{10} C_0\right)^2-\left({ }^{10} C_1\right)^2+\ldots-\left({ }^{10} C_9\right)^2+\left({ }^{10} C_{10}\right)^2

a

d

\left( {n + 1} \right)

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

(1-x)^{201}\left(1+x+x^2\right)^{200}

\left(\frac{1}{x}+2 x\right)^n

{x^5}

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

\left(\begin{array}{c}2 n+1 \\ 0\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 3\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 6\end{array}\right)+\ldots=170

n