Understanding Right Triangles and Integer Constraints

The video tutorial covers two math problems. Problem 248 involves constructing a right triangle on the xy plane with integer coordinates, and calculating the number of possible triangles. Problem 249 focuses on finding the number of prime factors of a number n, given certain conditions. The instructor explains each step in detail, providing insights into solving these types of problems efficiently.

What is the range of x-coordinates for point p in the right triangle problem?

• From -5 to 5

• From -4 to 5

• From -3 to 4

• From -4 to 4

How many different y-coordinates can point p have?

• 12

• 9

• 11

• 10

Once point p is placed, how many possible x-coordinates can point r have?

• 10

• 9

• 11

• 8

What must be true about the y-coordinate of point r?

• It must be the same as p's y-coordinate

• It must be less than p's y-coordinate

• It must be greater than p's y-coordinate

• It must be different from p's y-coordinate

How many possible y-coordinates can point q have, given a fixed position for p?

• 9

• 10

• 12

• 11

What is the total number of possible triangles that can be constructed?

• 900

• 1000

• 1100

• 990

In the new problem, what is the condition for n to be an integer?

• n must be greater than 200

• n must be less than 100

• n must be a prime number

• 14n must be divisible by 60

What is the smallest value of n that satisfies the condition in the new problem?

• 50

• 20

• 40

• 30

How many positive prime factors does 30 have?

• 2

• 5

• 3

• 4

If n is 60, how many positive prime factors does it have?

• 5

• 4

• 3

• 2