Both the roots of the equation x2 – Kx + 72 = 0 are integers. How many values are possible for K?6121824

The roots of the equation x^2 - Kx + 72 = 0 are integers. This means that the roots are factors of 72.

The factors of 72 are: ±1, ±2, ±3, ±4, ±6, ±8, ±9, ±12, ±18, ±24, ±36, ±72.

Each pair of factors will give a different value of K, because K is the sum of the roots (with a negative sign because of the equation's form).

For example, the factors 1 and 72 give K = -(1+72) = -73. The factors 2 and 36 give K = -(2+36) = -38.

So, we need to count the pairs of factors. There are 12 pairs of factors (including the negative ones), so there are 12 possible values for K.

However, we must remember that the question asks for the number of possible values for K, not the number of pairs of factors. Each pair of factors gives two possible values for K (one positive and one negative), so the total number of possible values for K is 2*12 = 24.

So, the answer is 24.