https://file.aiquickdraw.com/mathbot/file/1728741830798voeo61ku.jpg

Let's break down and solve the problems shown in the image.

### Problem 1: Find a Pythagorean triplet in which one member is 12.
A Pythagorean triplet consists of three integers \(a\), \(b\), and \(c\) that satisfy the equation:
\[
a^2 + b^2 = c^2
\]
When one of the members is 12, we can use the formula for generating triplets:
\[
a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2
\]
where \(m > n\) are positive integers. Let's substitute values to find a triplet containing 12.

For \(m = 2\) and \(n = 1\):
\[
a = 2^2 - 1^2 = 4 - 1 = 3, \quad b = 2(2)(1) = 4, \quad c = 2^2 + 1^2 = 4 + 1 = 5
\]
So the triplet is \( (3, 4, 5) \).

Now, we can scale this triplet by multiplying by a factor. To get 12, multiply the entire triplet by 3:
\[
(3 \times 3, 4 \times 3, 5 \times 3) = (9, 12, 15)
\]
Thus, the required triplet is \( (9, 12, 15) \).

---

### Problem 2: Find the smallest number by which 256 must be multiplied to obtain a perfect cube.
To find the smallest number that makes 256 a perfect cube, we start by performing its prime factorization:
\[
256 = 2^8
\]
For a number to be a perfect cube, the exponents in its prime factorization must be multiples of 3. Currently, \(2^8\) has an exponent of 8, which is not a multiple of 3. The smallest multiple of 3 greater than 8 is 9.

Thus, we need to multiply 256 by \(2^1\) (since \(9 - 8 = 1\)) to make the exponent of 2 equal to 9. Therefore, the smallest number by which 256 must be multiplied is:
\[
\boxed{2}
\]

---

### Problem 3: The adjoining figure RUNS is a parallelogram. Find \(x\) and \(y\) (lengths in cms).

Given the diagram of the parallelogram RUNS:
- \( RM = 10 \)
- \( SN = 10 \)
- \( UM = x + y \)
- \( UN = y + 4 \)
We are tasked with finding the values of \(x\) and \(y\).

Using properties of parallelograms:
1. Opposite sides are equal. Therefore, \(RM = SN\), so we have:
   \[
   10 = 10
   \]
   This does not provide new information.

2. For the diagonals, in a parallelogram, the diagonals bisect each other. This implies:
   \[
   UM = UN
   \]
   Substituting the given expressions:
   \[
   x + y = y + 4
   \]
   Solving for \(x\):
   \[
   x = 4
   \]
   Now substitute \(x = 4\) into \(UM = x + y\):
   \[
   10 = 4 + y
   \]
   Solving for \(y\):
   \[
   y = 6
   \]

Thus, the values of \(x\) and \(y\) are:
\[
x = 4 \quad \text{and} \quad y = 6
\]

---

Would you like more details or explanations on any of these? Here are 5 related questions to expand your understanding:

1. How can we generate other Pythagorean triplets using different values of \(m\) and \(n\)?
2. Why do we focus on prime factorization when finding the number that makes a given number a perfect cube?
3. What are the properties of parallelograms that help in solving geometric problems like this?
4. How can we solve for variables when given relationships between the diagonals in a parallelogram?
5. What are other real-life applications of Pythagorean triplets?

**Tip**: In geometric problems, always remember the key properties of the shapes involved (e.g., bisecting diagonals in parallelograms) to simplify the solution!

This page explains how to find a Pythagorean triplet where one number is 12, how to find the smallest multiplier to make 256 a perfect cube, and how to calculate the lengths of diagonals in a parallelogram. These problems involve key mathematical concepts such as Pythagoras' theorem, prime factorization, and geometric properties of parallelograms.

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