How many integral values of n satisfy the equation 2nx − 60 = 9x, where x is a natural number?

Step 1: First, we need to rearrange the equation to isolate n. We can do this by subtracting 60 from both sides and then dividing by 2x. This gives us:

n = (9x + 60) / 2x

Step 2: Now, we need to find the values of x for which n is an integer. Since x is a natural number, it can take any positive integer value.

Step 3: We can start by plugging in values for x. If x = 1, then n = 69/2, which is not an integer. If x = 2, then n = 34, which is an integer.

Step 4: We can continue this process, but it would be more efficient to find a pattern or rule. Notice that the numerator of the fraction is always increasing by 9, while the denominator is always increasing by 2. This means that n will be an integer whenever the numerator is an even number.

Step 5: The numerator is an even number when x is a multiple of 2. So, the integral values of n are those for which x is a multiple of 2.

Step 6: Since there is no upper limit on the values that x can take, there are infinitely many integral values of n that satisfy the equation.