RedDragonl

Analyzing the Problem

We have a quadrilateral BCED with extended sides meeting at point A. We're given several side lengths and need to find DE.

Solution Approach

To find DE, we'll need to break down the quadrilateral into triangles and use similar triangles to establish proportions.

Breaking Down the Quadrilateral

Let's divide quadrilateral BCED into two triangles: Triangle ABC and Triangle ADE.

Using Similar Triangles

We can see that triangle ABC and triangle ADE share angle A. Additionally, angle ABC is congruent to angle ADE (alternate interior angles since BD is parallel to AE). Therefore, triangles ABC and ADE are similar.

Setting up Proportions

Since the triangles are similar, their corresponding sides are proportional. We can set up the following proportion:

AB / BD = AC / DE

Substituting Values and Solving

We know AB = 3, BD = 18, and AC = 7. Substituting these values into the proportion, we get:

3 / 18 = 7 / DE

Cross-multiplying, we get:

3 * DE = 18 * 7

DE = (18 * 7) / 3

DE = 42

Therefore, DE = 42.

The answer is phi(1200) = 600.

The answer is phi(191) = 172.

The smallest number that works is 16.

The number we're looking at is interesting because it can be expressed as a difference of squares:

1,005,010,010,005,001 = (1005^2) - (1^2)

We can use a property of difference of squares to our advantage: a^2 - b^2 = (a + b)(a - b). In this case:

(1005^2) - (1^2) = (1005 + 1)(1005 - 1) = 1006 * 1004

Now, 1006 is even (because the last digit is 6), so it's not prime. We can focus on 1004.

Looking closely at 1004, we see that the last two digits are 04, which is a multiple of 4. Therefore, 1004 is also even and not prime.

However, 1004 can be further factored: 1004 = 2 * 2 * 251. Here, 251 is odd, and a quick primality check reveals it is a prime number.

Therefore, the largest prime factor of 1,005,010,010,005,001 is 251.

The answer to the problem is sqrt(17).

For

x+y+z=17, 

xy+xz+yz=94,

xyz=168

you can look at all the factors of 168