In this problem, we prove a property of integers expressed in algebraic form. We are required to show that the square of any positive integer of the form 5q + 1 is again of the same form.
Question
Prove that the square of any positive integer of the form 5q + 1 is of the same form.
Solution
Let the given positive integer be of the form
5q + 1, where q is an integer.
Now, square the given number:
(5q + 1)²
= 25q² + 10q + 1
Rewrite the expression as:
25q² + 10q + 1
= 5(5q² + 2q) + 1
Since q is an integer, the expression (5q² + 2q) is also an integer.
Therefore, the square of the given number can be written in the form
5k + 1, where k is an integer.
Hence, the square of any positive integer of the form 5q + 1 is again of the same form.
Conclusion
Thus, the square of any positive integer of the form 5q + 1 is also of the form 5q + 1.
Hence proved.