Let the three consecutive odd integers be represented as $x$, $x+2$, and $x+4$.

The sum of these three integers is given as:

$x + (x + 2) + (x + 4) = 153$

Simplify the equation:

$3x + 6 = 153$

Subtract 6 from both sides:

$3x = 147$

Now, divide both sides by 3:

$x = 49$

So, the three consecutive odd integers are $49$, $51$, and $53$. The largest integer is:

$\boxed{53}$

Would you like a detailed breakdown or have any other questions?

Here are some related questions you can explore:

1. What if the sum was 195? What would the integers be?
2. How would the solution change if they were consecutive even integers?
3. What happens if you add four consecutive odd integers instead of three?
4. How can this method be applied to find five consecutive odd integers with a given sum?
5. How does solving for consecutive integers differ from solving for multiples of integers?

Tip: When dealing with consecutive odd or even integers, always represent them as $x$, $x+2$, $x+4$, etc.