Is it always true that $\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$?

Question

My question is pretty basic.

Here it goes:

Is it always true that $$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1 \pmod 4$$ where the $s_j$'s are positive integers, and may be odd or even?

We can perhaps assume that $w \geq 10$.

I am thinking that it might be possible to disprove this, but I cannot think of a counterexample at this moment.

MY ATTEMPT

How about $$w=10$$ $$s_1 = s_2 = s_3 = s_4 = s_5 = s_6 = s_7 = s_8 = s_9 = 2$$ and $$s_{10} = 1$$ for a counterexample?

Answer 1

Each factor on the left hand side may be either $\equiv 1$ or $\equiv 3\bmod 4$. If there are an odd number of the latter (i.e. an odd number of odd $s_j$), the product will be $\equiv 3\bmod 4$.

Answer 2

Since $2x+1\equiv(-1)^x\pmod 4$, we have $$\prod_{j=1}^{w}{(2s_j + 1)} \equiv (-1)^{\sum_{j}s_j} \pmod 4$$ hence $$\prod_{j=1}^{w}{(2s_j + 1)} \equiv 1\pmod 4\iff\sum_{j=1}^ws_j\equiv 0\pmod 2$$