Let m be a natural number (1<=m<=100). For how many values of m, 4m+1 is a perfect square.
Answers
Answer:
Step-by-step explanation:
Our aim is to find for how many values of m, the given expression is a perfect square.
In order to be a perfect square it has to be expressed as a multiplication of two equal numbers.
As an example consider: 9 = 3 x 3, then 9 is a perfect square.
We know that m is less or equal to 100, hence 4m + 1 will be less or equal to 401.
Lets consider the case when:
- m = 2:
4 x 2 + 1 = 9, is a perfect square.
- m = 6:
4 x 6 + 1 = 25, is a perfect square.
- m = 12:
4 x 12 + 1 = 49, is a perfect square.
- m = 20:
4 x 20 + 1 = 81, is a perfect square.
- m = 30:
4 x 30 + 1 = 121, is a perfect square.
- m = 42:
4 x 42 + 1 = 169, is a perfect square.
- m = 56:
4 x 56 + 1 = 225, is a perfect square.
- m = 72:
4 x 72 + 1 = 289, is a perfect square.
- m = 90:
4 x 90 + 1 = 361, is a perfect square.
Hence, 4m + 1 is a perfect square for 9 values of m, (2, 6, 12, 20, 30, 42, 56, 72, 90).