6. If M is replaced by one of the following numbers, the product \( (M-10)(M+14) \) becomes a perfect square. A. 0 B. 12 C. 14 D. 16 E. 18 7. In a triangle the three angles are in the ratio of \( 2: 3: 5 \). The difference between the largest and the smallest angle, in degrees = ? A. 0 B. 120 C. 90 D. 54 E. 72

**Problem 6** We are given the expression \[ (M-10)(M+14) \] and we need to find which value of \( M \) makes this product a perfect square. We test the choices: - **A. \( M=0 \)** \[ (0-10)(0+14) = (-10)(14) = -140 \] A negative number is not a perfect square (in the set of nonnegative integers). - **B. \( M=12 \)** \[ (12-10)(12+14) = (2)(26) = 52 \] \( 52 \) is not a perfect square. - **C. \( M=14 \)** \[ (14-10)(14+14) = (4)(28) = 112 \] \( 112 \) is not a perfect square. - **D. \( M=16 \)** \[ (16-10)(16+14) = (6)(30) = 180 \] \( 180 \) is not a perfect square. - **E. \( M=18 \)** \[ (18-10)(18+14) = (8)(32) = 256 \] \( 256 \) is equal to \( 16^2 \), which is a perfect square. Thus, the answer for Problem 6 is **E**. --- **Problem 7** The angles of a triangle are in the ratio \( 2:3:5 \). Let the measures of the angles be: \[ 2x, \quad 3x, \quad 5x \] Since the sum of the angles in a triangle is \( 180^\circ \), we have: \[ 2x + 3x + 5x = 180 \] \[ 10x = 180 \] \[ x = 18 \] Now, calculate each angle: \[ \text{Smallest angle} = 2x = 2(18) = 36^\circ \] \[ \text{Largest angle} = 5x = 5(18) = 90^\circ \] The difference between the largest and the smallest angle is: \[ 90^\circ - 36^\circ = 54^\circ \] Thus, the answer for Problem 7 is **54**, which corresponds to option **D**.