Sum of consecutive integers (ACT). Game on.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 75$, so $3n + 6 = 75$, $3n = 69$, and $n = 23$. The largest is $n + 4 = 27$.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 84$, so $3n + 6 = 84$, $3n = 78$, and $n = 26$. The largest is $n + 4 = 30$.

Let the integers be $n$, $n + 1$, and $n + 2$. Then $n + (n + 1) + (n + 2) = 51$, so $3n + 3 = 51$, $3n = 48$, and $n = 16$. The smallest is $16$.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 63$, so $3n + 6 = 63$, $3n = 57$, and $n = 19$. The largest is $n + 4 = 23$.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 78$, so $3n + 6 = 78$, $3n = 72$, and $n = 24$. The smallest is $24$.

Let the integers be $n$, $n + 1$, and $n + 2$. Then $n + (n + 1) + (n + 2) = 90$, so $3n + 3 = 90$, $3n = 87$, and $n = 29$. The largest is $n + 2 = 31$.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 105$, so $3n + 6 = 105$, $3n = 99$, and $n = 33$. The middle is $n + 2 = 35$.

Let the integers be $n$, $n + 2$, and $n + 4$. Then $n + (n + 2) + (n + 4) = 99$, so $3n + 6 = 99$, $3n = 93$, and $n = 31$. The smallest is $31$.