if $xy=2808, z(y+4)=2808$ find $y$

Question

Let $x,y,z$ are postive integer,and such $$\begin{cases} x\ge 30\\ xy=2808\\ z(y+4)=2808 \end{cases}$$ Find $y$

I have take $2808=2\times 1404=2^3\times 3^3\times 13$

Answer 1

Once you know $y$, then you easily get $x$ and $z$. There are 3 conditions $y$ must satisfy: $y \mid 2808$, $(y+4) \mid 2808$ and $y\leq 2808/30=93.6.$ So write down all the divisors of $2808$ in increasing order and find all pairs which differ by $4$. This gives you all the candidates for $y$. Eliminate all $y$'s which are bigger than $93.6$.

To be a little more efficient, note that $y(y+4)\leq 2808$, which implies that $y\leq 51$.

Answer 2

Since $y = \frac{2808}{x}$ and $x \geq 30$, we have that $y \leq \frac{2808}{30} = 93.6 \implies y \leq 93$.

Since $y$ divides $2808$, $y = 2^a \cdot 3^b$ or $y = 2^a \cdot 3^b \cdot 13$ for $0 \leq a, b \leq 3$.

Writing down such $y \leq 93$ sequentially is trivial but tedious, and gives the list: $y \in \lbrace 1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 27, 36, 52, 54, 72, 78 \rbrace$

Observe that $z(y+4) = 2808 \implies y+4$ divides $2808$, and that the only $y$ in the list such that $y+4$ dividies 2808 are $y = 2, 4, 8, 9$.

Then $x = \frac{2808}{y}$ and $z = \frac{2808}{y+4}$ gives solutions $$(x, y, z) = (1404, 2, 468), \;(702, 4, 351), \; (351, 8, 234), \;(312, 9, 216)$$

Answer 3

Hint : Find positive divisors of $2808$ with difference $4$. $y$ is the smaller of those divisors. Make sure that $y$ is not exceeding $93$