The GCD of three numbers is 30 and their LCM is 900. Two of the numbers are 60 and 150. Find the least possible value of the third number.

Let the third number be $C = 2^x \times 3^y \times 5^z$.

For GCD, the exponent of each prime factor is the minimum of its exponents in the numbers.

This implies:

• $\min(2, 1, x) = 1 \implies x \ge 1$
• $\min(1, 1, y) = 1 \implies y \ge 1$
• $\min(1, 2, z) = 1 \implies z \ge 1$

For LCM, the exponent of each prime factor is the maximum of its exponents in the numbers.

This implies:

• $\max(2, 1, x) = 2 \implies x \le 2$
• $\max(1, 1, y) = 2 \implies y = 2$ (since $y \ge 1$)
• $\max(1, 2, z) = 2 \implies z = 2$ (since $z \ge 1$)

To find the least possible value of $C$, we choose the smallest possible exponents for $x, y, z$ that satisfy all conditions.

• For $x$: $x \ge 1$ and $x \le 2$. The smallest possible integer value is $x=1$.
• For $y$: $y=2$.
• For $z$: $z=2$. So, $C = 2^1 \times 3^2 \times 5^2$.

The least possible value of the third number is $\boxed{\text{450}}$.

Substitute these into the equation:

The value of $x$ is $\boxed{\frac{1}{2}}$.

The sum of interior angles of a polygon with $k$ sides is $S_k = (k-2) \times 180^\circ$.

For the polygon with $n$ sides: $S_n = (n-2) \times 180^\circ$.
For the polygon with $n+2$ sides: $S_{n+2} = ((n+2)-2) \times 180^\circ = n \times 180^\circ$.
The ratio $S_n : S_{n+2} = 3:4$ means:

Cancel out $180^\circ$ from the numerator and denominator:

Cross-multiply:

The value of $n$ is $\boxed{8}$.

The polygon has $n=8$ sides.
The formula for each exterior angle of a regular polygon with $k$ sides is $E_k = \frac{360^\circ}{k}$.
For the polygon with $n=8$ sides:

The size of each exterior angle of the polygon with $n$ sides is $\boxed{45^\circ}$.