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### Problem 19:
**Find the smallest number by which 256 must be multiplied to obtain a perfect cube.**

To solve this, we start by finding the prime factorization of 256:
\[
256 = 2^8
\]
To form a perfect cube, the powers of all prime factors must be multiples of 3. Since \( 2^8 \) has a power of 8, we need to multiply by \( 2^1 \) to make the exponent a multiple of 3:
\[
8 + 1 = 9 \quad \text{(which is divisible by 3)}
\]
Thus, we need to multiply 256 by \( 2^1 = 2 \).

**Answer:** The smallest number to multiply by is **2**.

---

### Problem 20:
**In the parallelogram RUNS, find the values of x and y (lengths in cm).**

The parallelogram has a diagonal marked with values involving \(x\) and \(y\):
- \( RN = x + y \)
- \( UN = y + 4 \)

It is given that diagonals of a parallelogram bisect each other, meaning the lengths of \( RN \) and \( UN \) will be equal:
\[
x + y = y + 4
\]

Now, subtract \(y\) from both sides:
\[
x = 4
\]

Once we have \(x = 4\), substitute it into the equation \(RN = x + y\):
\[
4 + y = y + 4
\]

Thus, the value of \(x\) is **4** and you can find \(y\) by solving the equations or determining it's a unique solution from symmetry, but with all lengths in cm.

---

Would you like more details on any of these problems? Here are some related questions for further practice:

1. What is the prime factorization of 512?
2. Can you explain why diagonals of a parallelogram bisect each other?
3. What are the properties of a perfect square compared to a perfect cube?
4. How would you find the cube root of 4096?
5. What happens when the diagonals of a parallelogram are not equal?

**Tip:** When solving geometry problems, always consider the properties of the shape, such as symmetry or equal sides, as they often provide clues to the solution.

1. Find the smallest number by which 256 must be multiplied to obtain a perfect cube. 2. In the parallelogram RUNS, find the values of x and y (lengths in cm).

Learn how to find the smallest number by which 256 must be multiplied to become a perfect cube and how to determine x and y in a parallelogram using properties of bisected diagonals. Suitable for students in Grades 8-10.

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