Integer Square Condition Setup

Let the positive integer be represented by '$n$'. The problem states that the square of the integer ('$n^2$') is greater than 5 times the integer ('$5n$') by $-6$. This translates to the equation:

$n^2 = 5n + (-6)$

$n^2 = 5n - 6$

Solving the Quadratic Equation

To find the integer '$n$', we rearrange the equation into the standard quadratic form ($ax^2 + bx + c = 0$):

$n^2 - 5n + 6 = 0$

Now, we factor the quadratic expression:

$ (n - 2)(n - 3) = 0 $

This equation yields two possible integer solutions:

• $n - 2 = 0 \implies n = 2$
• $n - 3 = 0 \implies n = 3$

Identifying the Least Positive Integer

The possible integer values for '$n$' are 2 and 3. Both are positive integers.

The question asks for the least positive integer that satisfies the condition.

Comparing the solutions {2, 3}, the least value is 2.

Final Answer

The least positive integer satisfying the condition is 2.