Animated Solution for Mathematics - Binomial Theorem: $25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by

Imagine you are standing before a massive, intimidating wall of numbers: $N=25^{190}-19^{190}-8^{190}+2^{190}$. It looks like a monster, but in JEE Advanced mathematics, these problems are tests of your ability to see hidden order within chaos.

Our mission is to determine the divisibility of $N$ by $14$ and $34$.

The first step in taming this beast is to recognize a pattern. We have four terms that are begging to be paired.

Recalling the algebraic identity that $a^n-b^n$ is always divisible by $(a-b)$, we rearrange $N$ as follows:

In the first bracket, $25^{190}-8^{190}$ is divisible by $25-8=17$. In the second bracket, $19^{190}-2^{190}$ is divisible by $19-2=17$.

Since both parts are multiples of $17$, their difference $N$ must also be a multiple of $17$.

To check for divisibility by $34$, we note that $34=17\times 2$. We have already established that $N$ is divisible by $17$.

Now, we check if $N$ is even. Observe the parity of the bases: $25$ and $19$ are odd, while $8$ and $2$ are even.

Since any power of an odd number is odd and any power of an even number is even, we have:

An odd minus an odd is even, and adding or subtracting even numbers preserves this parity. Thus, $N$ is even, meaning it is divisible by $2$.

Since $N$ is divisible by both $17$ and $2$, and these factors are coprime, $N$ is definitely divisible by $34$.

Finally, we test for divisibility by $14$. Since we know $N$ is divisible by $2$, we only need to check if it is divisible by $7$.

We reduce each base modulo $7$: $25\equiv 4(\mathrm{mod}7)$ $19\equiv 5\equiv -2(\mathrm{mod}7)$ $8\equiv 1(\mathrm{mod}7)$ $2\equiv 2(\mathrm{mod}7)$

Substituting these into $N$, we get:

Since $190$ is even, $(-2{)}^{190}=2^{190}$. The expression simplifies to:

We know $4^3=64\equiv 1(\mathrm{mod}7)$. Since $190=3\times 63+1$, we have:

Finally, $N\equiv 4-1=3(\mathrm{mod}7)$. Because the remainder is $3$, $N$ is not divisible by $7$.

Conclusion: $N$ is divisible by $34$ but not by $14$.