A die was rolled a number of times. The outcomes are: Number 1 2 3 4 Outcomes 30 26 20 23

Total rolls = $30 + 26 + 20 + K + 23 + 24 = 123 + K$.

The number of times 1 was obtained is 30.

The value of K is $\boxed{27}$.

Even numbers are 2, 4, 6.
Frequency of 2 = 26
Frequency of 4 = K = 27
Frequency of 6 = 24
Number of even outcomes = $26 + 27 + 24 = 77$.

Factors of 6 are 1, 2, 3, 6.
Frequency of 1 = 30
Frequency of 2 = 26
Frequency of 3 = 20
Frequency of 6 = 24
Number of factor of 6 outcomes = $30 + 26 + 20 + 24 = 100$.

These are 2 and 6.
Number of (even AND factor of 6) outcomes = $26 + 24 = 50$.

The probability is $\boxed{\frac{127}{150}}$.

Multiply the entire inequality by 30.

The solution is $\boxed{x \ge -\frac{1}{9}}$.

:
In the diagram, TU is a tangent to the circle. $\angle RVU = 100^\circ$, and $\angle ZUV = 36^\circ$. Find the value of $\angle ZSTU$.
(Assuming the points Z, S, V, U are on the circle, and R is also on the circle, forming a cyclic pentagon ZSVRU. The angle $\angle ZSTU$ is interpreted as $\angle ZSU$.)

Given $\angle ZUV = 36^\circ$. This angle is subtended by arc ZV at point U on the circumference.
Therefore, the angle subtended by the same arc ZV at point S on the circumference is also $36^\circ$.

The angle between the tangent TU and the chord UV is $\angle TUV$. This angle is equal to the angle subtended by the chord UV in the alternate segment, which is $\angle USV$.
So, $\angle TUV = \angle USV$.

Since Z, S, V, R, U are points on the circle, $\angle RVU$ is an angle subtended by arc RU at V.
Therefore, $\angle RSU = \angle RVU = 100^\circ$ (angles subtended by the same arc RU).

We know $\angle RSU = 100^\circ$.
We can write $\angle RSU = \angle RSZ + \angle ZSU$.
However, we need $\angle ZSU$. We have $\angle ZSV = 36^\circ$.
We also know $\angle RSU = \angle RSV + \angle VSU$.
From the diagram, it appears that Z, S, V, R are consecutive points on the circle.
If $\angle RSU = 100^\circ$ and $\angle ZSV = 36^\circ$.
We need $\angle ZSU$.
From the diagram, $\angle ZSU = \angle RSU - \angle RZ_1U$ (This is not correct).
Let's use the tangent-chord theorem for $\angle TUZ$.
$\angle TUZ = \angle USZ$.
We have $\angle TUV = \angle USV$.
We know $\angle ZUV = 36^\circ$.
We know $\angle RVU = 100^\circ$.
From the diagram, $\angle TUV = \angle TUZ + \angle ZUV$. This is incorrect. $\angle TUV$ is the angle between tangent TU and chord UV.
$\angle TUZ$ is the angle between tangent TU and chord UZ.
$\angle ZUV$ is an angle inside the circle.

Let's re-evaluate the interpretation of $\angle ZSTU$. It is likely $\angle ZSU$.
We have $\angle ZSV = 36^\circ$.
We have $\angle RSU = 100^\circ$.
From the diagram, it looks like Z, S, V, R are in order around the circle.
Then $\angle ZSU = \angle ZSV + \angle VSU$.
We need $\angle VSU$.
By the alternate segment theorem, $\angle VSU = \angle TUV$.
We know $\angle ZUV = 36^\circ$.
We know $\angle RVU = 100^\circ$.
The angle $\angle TUV$ is the angle between the tangent TU and the chord UV.
The angle $\angle TUZ$ is the angle between the tangent TU and the chord UZ.
The angle $\angle TUV$ is not directly given.

Let's assume the angle $100^\circ$ is $\angle ZRV$.
If Z, R, V, U are points on the circle, then $\angle ZRV + \angle ZUV = 180^\circ$.
So $100^\circ + 36^\circ = 136^\circ \ne 180^\circ$. So ZRVU is not a cyclic quadrilateral.

Let's assume the angle $100^\circ$ is $\angle VUR$. This is the angle between the tangent TU and the chord UV.
If $\angle VUR = 100^\circ$, then by the alternate segment theorem, $\angle USV = 100^\circ$.
We are given $\angle ZUV = 36^\circ$.
Since Z, S, V, U are on the circle, $\angle ZSV = \angle ZUV = 36^\circ$.
We need to find $\angle ZSU$.
$\angle ZSU = \angle ZSV + \angle VSU$.

This interpretation seems plausible. Let's assume $\angle ZRVU = 100^\circ$ means $\angle VUR = 100^\circ$.

Given that TU is a tangent at U, and UV is a chord. The angle between the tangent and the chord, $\angle TUV$, is equal to the angle subtended by the chord in the alternate segment, $\angle USV$.
From the problem statement, $\angle ZRVU = 100^\circ$. Given the diagram, this is most likely referring to $\angle TUV = 100^\circ$ (the angle between the tangent and the chord UV).
So, $\angle USV = 100^\circ$.

Given $\angle ZUV = 36^\circ$. This angle is subtended by arc ZV at point U on the circumference.
The angle subtended by the same arc ZV at point S on the circumference is $\angle ZSV$.
Therefore, $\angle ZSV = \angle ZUV = 36^\circ$.

From the diagram, $\angle ZSU = \angle ZSV + \angle VSU$.

The value of $\angle ZSTU$ (interpreted as $\angle ZSU$) is $\boxed{136^\circ}$.

:
Given that $(7-2x)$, 9, $(5x+17)$ are consecutive terms of a Geometric Progression with common ratio $r>0$. Find the values of x.

In a GP, the ratio of consecutive terms is constant. So, $\frac{9}{7-2x} = \frac{5x+17}{9}$.

Here, $a=10$, $b=-1$, $c=-38$.

For $x=2$:
The terms are $(7-2(2))$, 9, $(5(2)+17)$.
Terms are $(7-4)$, 9, $(10+17)$.
Terms are $3, 9, 27$.
The common ratio $r = \frac{9}{3} = 3$. Since $r=3 > 0$, $x=2$ is a valid solution.
For $x=-1.9$:
The terms are $(7-2(-1.9))$, 9, $(5(-1.9)+17)$.
Terms are $(7+3.8)$, 9, $(-9.5+17)$.
Terms are $10.8, 9, 7.5$.
The common ratio $r = \frac{9}{10.8} = \frac{90}{108} = \frac{5}{6}$. Since $r=\frac{5}{6} > 0$, $x=-1.9$ is also a valid solution.
The problem asks for "the values of x", implying both are valid.

The values of x are $\boxed{2 \text{ and } -1.9}$.

:
Two positive numbers are in the ratio 3:4. The sum of thrice the first number and twice the second is 68. Find the smaller number.

Let the two positive numbers be $3k$ and $4k$, where $k$ is a positive constant.
The first number is $3k$. The second number is $4k$.

Thrice the first number = $3 \times (3k) = 9k$.
Twice the second number = $2 \times (4k) = 8k$.
The sum is 68:

First number = $3k = 3 \times 4 = 12$.
Second number = $4k = 4 \times 4 = 16$.

The smaller number is 12.
The smaller number is $\boxed{12}$.