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

In order for the expression \( (M-10)(M+14) \) to be a perfect square, we can substitute the given options for \( M \) and check the results. For instance, if we take \( M = 0 \), then the product becomes \( (-10)(14) = -140 \), which is not a perfect square. However, substituting \( M = 16 \) yields \( (16-10)(16+14) = 6 \times 30 = 180 \), once again not a perfect square. Playing around with values reveals \( M = 12 \) results in \( (12-10)(12+14) = 2 \times 26 = 52 \), leading to the conclusion that only \( M = 18 \) works since it perfectly squares \( 64 \). Triangles can be an interesting adventure in angles! Given the ratio of \( 2:3:5 \), we can let the angles be \( 2x, 3x, \) and \( 5x \). The total of angles in a triangle equals \( 180^\circ \). Setting up an equation: \( 2x + 3x + 5x = 180 \) gives us \( 10x = 180 \) or \( x = 18 \). Therefore, the angles are \(36^\circ, 54^\circ, \) and \(90^\circ\). The difference between the largest angle \(90^\circ\) and the smallest angle \(36^\circ\) is \(54^\circ\)!