## To find the next larger integer that is both a square and a cube of a positive integer, we can use the factoring method mentioned above.

First, we need to find an integer that satisfies the conditions of being both a square and a cube, which means that all of its prime exponents must be divisible by 6.

Starting with the smallest possible value, which is 1, let's check if it satisfies the conditions. Since the exponent of 2 is 0, which is divisible by 6, and there are no other prime factors, 1 is indeed both a square and a cube.

Now, let's find the next larger integer. We can choose a = 6 and b = c = ... = 0 to obtain an integer that satisfies the conditions. Plugging these values into the factoring formula, we get:

n = 2^(6) * 3^(0) * 5^(0) * ...

Evaluating this expression, we find that n is equal to 64. Therefore, the next larger integer that is both a square and a cube is 64.

So, in conclusion, the next larger integer that is both a square and a cube of a positive integer is 64.