Let $$𝑁$$ be the coefficient of $$x^{2025}$$ in the expansion of $$(x+1)(x^{2}+3)(x^{4}+5)(x^{8}+7)....(x^{1024}+21)$$. What is the remainder when $$𝑁$$ is divided by $$100$$?

The expansion is

$$(x+1)(x^{2}+3)(x^{4}+5)(x^{8}+7)\dotsm(x^{1024}+21).$$

There are $$11$$ factors. In general, the $$k^{\text{th}}$$ factor is
$$(x^{2^{k}} + (2k+1)), \qquad k = 0,1,2,\dots ,10.$$

While forming a particular term we either pick $$x^{2^{k}}$$ (with coefficient $$1$$) or the constant $$(2k+1)$$ from the $$k^{\text{th}}$$ factor. Hence:

• If we pick $$x^{2^{k}}$$ from a factor, the power $$2^{k}$$ is added to the total exponent.
• If we pick the constant, the factor contributes a multiplicative constant $$(2k+1)$$ to the coefficient.

Thus, to obtain $$x^{2025}$$ the set of chosen exponents must satisfy

$$\sum_{k\in S} 2^{k} = 2025,$$

where $$S$$ is the set of indices from which we selected $$x^{2^{k}}$$. Because powers of two are unique, this representation is the binary expansion of $$2025$$.

Write $$2025$$ in binary:

$$2025 = 1024 + 512 + 256 + 128 + 64 + 32 + 8 + 1$$

Corresponding indices:

$$S = \{10,\,9,\,8,\,7,\,6,\,5,\,3,\,0\}.$$

The remaining indices $$\{1,2,4\}$$ are those from which we must take the constant terms. Therefore the required coefficient is

$$N = (2\cdot 1 + 1)\,(2\cdot 2 + 1)\,(2\cdot 4 + 1) = 3 \times 5 \times 9 = 135.$$

We need the remainder of $$N$$ modulo $$100$$:

$$135 \equiv 35 \pmod{100}.$$

Hence the remainder is 35.