  The query model has generated considerable interest in both classical and
quantum computing communities. Typically, quantum advantages are demonstrated
by showcasing a quantum algorithm with a better query complexity compared to
its classical counterpart. Exact quantum query algorithms play a pivotal role
in developing quantum algorithms. For example, the Deutsch-Jozsa algorithm
demonstrated exponential quantum advantages over classical deterministic
algorithms. As an important complexity measure, exact quantum query complexity
describes the minimum number of queries required to solve a specific problem
exactly using a quantum algorithm.
  In this paper, we consider the exact quantum query complexity of the
following two $n$-bit symmetric functions $\text{MOD}_m^n:\{0,1\}^n \rightarrow
\{0,...,m-1\}$ and $\text{EXACT}_{k,l}^n:\{0,1\}^n \rightarrow \{0,1\}$, which
are defined as $\text{MOD}_m^n(x) = |x| \bmod m$ and $ \text{EXACT}_{k,l}^n(x)
= 1$ iff $|x| \in \{k,l\}$, where $|x|$ is the number of $1$'s in $x$. Our
results are as follows: i) We present an optimal quantum algorithm for
computing $\text{MOD}_m^n$, achieving a query complexity of $\lceil
n(1-\frac{1}{m}) \rceil$ for $1 < m \le n$. This settles a conjecture proposed
by Cornelissen, Mande, Ozols and de Wolf (2021). Based on this algorithm, we
show the exact quantum query complexity of a broad class of symmetric functions
that map $\{0,1\}^n$ to a finite set $X$ is less than $n$. ii) When $l-k \ge
2$, we give an optimal exact quantum query algorithm to compute
$\text{EXACT}_{k,l}^n$ for the case $k=0$ or $k=1,l=n-1$. This resolves the
conjecture proposed by Ambainis, Iraids and Nagaj (2017) partially.
