In this article, we have discussed two frequently asked questions of CAT and other MBA exams.

**Question 1: For a given perimeter how many triangles with integral sides are possible?**

Many of us try solving this manually. But with the help of a shortcut that is discussed in this article, we can solve this question within seconds.

To solve this question, we have to consider two cases, i.e.

Case 1: when Perimeter is even

Case 2: when Perimeter is odd

Let us discuss these cases in detail.

**Case 1: How many triangles with integral sides are possible for perimeter P where P is even**

In this case, total number of triangles will be the nearest integer to P^{2}/48

**Example:** How many triangles with integral sides are possible for perimeter = 18?

**Solution:** Nearest integer to (18^2)/48 is 7, so 7 such triangles are possible

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**Case 2: How many triangles with integral sides are possible for perimeter P where P is odd**

In this case, total number of triangles will be the nearest integer to (P+3)^{2}/48

**Example:** How many triangles with integral sides are possible for perimeter = 19?

**Solution:** Nearest integer to {(19+3)^2}/48 is 10, so 10 such triangles are possible.

**Question 2: How many right angled triangles with integral sides are possible with only one of the perpendicular sides given?**

Generally, we try solving this question using triplets and we tend to miss cases.

But with the help of the given trick, we can solve it in lesser time span.

Again, there can be two cases.

Case 1: when the perpendicular side is even

Case 2: when the perpendicular side is odd

**Case 1: How many right angled triangles with integral sides are possible with perpendicular P where P is even**

In this case, total number of triangles will be [(factors of (P^{2}/4)) – 1]/2

**Example:** How many right angled triangles with integral sides are possible with perpendicular side as 40?

**Solution:** Total number of factors of 40^{2}/4 i.e. 400 is 15, so (15-1)/2=7 such triangles are possible

**Case 2: How many right angled triangles with integral sides are possible with perpendicular P where P is odd**

In this case, total number of triangles will be equal to [(factors of P^{2}) – 1]/2

**Example:** How many right angle triangles with integral sides are possible with perpendicular side as 15?

**Solution:** Total number of factors of 15^{2} i.e. 225 is 9, so (9-1)/2=4 such triangles are possible

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