Question

How many natural numbers less than 1000 can be expressed as the difference of two perfect squares in at least one way?​

Answer 1

Question :- How many natural numbers which are also perfects square less than 1000 can be expressed as the difference of two perfect squares in at least one way ?

Answer :-

we need to find the natural numbers which are also perfect squares less than 1000.

Or we can see that, we need to find pythagorean triplets less than 1000.

As, pythagorean triplets are in the form of :-

• a² + b² = c² or, a² = c² - b².
• Difference of two square number is also a square number.

Now we have to check all the triplets whose square is less than 1000.

So, required natural numbers are :-

• 3² = 5² - 4² => 9 = 25 - 16 .
• 6² = 10² - 8² => 36 = 100 - 64.
• 5² = 13² - 12² => 25 = 169 - 144.
• 9² = 15² - 12² => 81 = 225 - 144 .
• 8² = 17² - 15² => 64 = 289 - 225 .
• 7² = 25² - 24² => 49 = 625 - 576 .
• 20² = 29² - 21² => 400 = 841 - 441.

Therefore,

→ the natural numbers that can be represented as a difference of squares and they are also perfect squares are = 9, 16, 25, 36, 49, 64, 81, 144, 225, 400, 441 and 576 .