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We discuss an elementary method for the evaluation of the convolution sums$\underset{\substack{  {(l,m)\in\mathbb{N}_{0}^{2}} \\ {\alpha\,l+\beta\,m=n} }}{\sum}\sigma(l)\sigma(m)$ for those $\alpha,\beta\in\mathbb{N}$ for which$\gcd{(\alpha,\beta)}=1$ and $\alpha\beta=2^{\nu}\mho$, where$\nu\in\{0,1,2,3\}$ and $\mho$ is a finite product of distinct odd primes.Modular forms are used to achieve this result. We also generalize theextraction of the convolution sum to all natural numbers. Formulae for thenumber of representations of a positive integer $n$ by octonary quadratic formsusing convolution sums belonging to this class are then determined when$\alpha\beta\equiv 0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. To achieve thisapplication, we first discuss a method to compute all pairs$(a,b),(c,d)\in\mathbb{N}^{2}$ necessary for the determination of such formulaefor the number of representations of a positive integer $n$ by octonaryquadratic forms when $\alpha\beta$ has the above form and $\alpha\beta\equiv0\pmod{4}$ or $\alpha\beta\equiv 0\pmod{3}$. We illustrate our approach byexplicitly evaluating the convolution sum for $\alpha\beta=33=3\cdot 11,\>\alpha\beta=40=2^{3}\cdot 5$ and $\alpha\beta=56=2^{3}\cdot 7$, and byrevisiting the evaluation of the convolution sums for $\alpha\beta=10$, $11$,$12$, $15$, $24$. We then apply these convolution sums to determine formulaefor the number of representations of a positive integer $n$ by octonaryquadratic forms. In addition, we determine formulae for the number ofrepresentations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(2,3)$,$(1,9)$.

Elementary Evaluation of Convolution Sums involving the Sum of Divisors Function for a Class of positive Integers