Solve the congruence x^329 = 452 (mod 1147). [hint. 1147 is not prime.]

The answer is

Step-by-step explanation:

The following steps will give a solution to the congruence

1. Compute Euler's Phi function .

We have by prime factorization, so that

because where p is a prime number.

2. Find positive integers u and v that satisfy .

We know a solution exists, since , using the Euclidean algorithm allows us to find the solution

In order to get positive values for u and v, we modify this solution:

and

The equation

provides the key to solving the original problem.

3. Compute by successive squaring. The value obtained gives the solution x.

We have , so

To use this method start by looking at the exponent 929 and represent it as a sum of powers of 2 this is called the binary expansion of 929. To do this, find the largest power of 2 less than your exponent in this case it’s . Subtract 512 from 929 getting 417. And continue in this manner to get:

Now

So all you have to do is to calculate the numbers

and multiply them together, then take the product

answer:   the paragraph proof is mentioned below.

step-by-step explanation:

given: in a δxwy where

such that, ∠wzx ≅ ∠yzx and zw≅zy

we have to prove that: segment zx is a perpendicular bisector of segment wy.

prove: since, zw≅zy ⇒ ∠wzy is a supplementary angle.

then, ∠wzy= 180°

⇒∠wzx + ∠yzx = 180° ( because xz divides ∠wzy into two parts)

but, ∠wzx ≅ ∠yzx ( given)

⇒∠wzy+∠wzy=180° ( by substitution)

⇒2∠wzy=180°

⇒∠wzy=90°

⇒∠wzy=∠yzx =90°

therefore, xz⊥wy where wz=zy

⇒zx is a perpendicular bisector of wy.