Problem 1. Determine the integers $a, b, c$ for which

$$
\frac{a+1}{3}=\frac{b+2}{4}=\frac{5}{c+3}
$$

Mathematical Gazette
## Solution and Grading Criteria

Since $(3,4)=1$, from $3(b+2)=4(a+1)$ it follows that $3 \mid a+1$, so the common value of the fractions in the statement is an integer. ......................................................... 2 points We deduce that $c+3 \mid 5$, so $c+3 \in\{-5,-1,1,5\}$, which means $c \in\{-8,-4,-2,2\} . \ldots \ldots \ldots 3$ points. In the case $c=-8$, we find $a=-4, b=-6$; in the case $c=-4$, we get $a=-16, b=-22$; in the case $c=-2$, we find $a=14$ and $b=18$, and in the case $c=2$, we get $a=b=2$. . . 2 points

Remark: Another way to show that the common value of the fractions in the statement is an integer is:

$\frac{a+1}{3}=\frac{b+2}{4}=\frac{(b+2)-(a+1)}{4-3}=b-a+1 \in \mathbb{Z}$.