Question

Some of the natural numbers can be written as the difference of two squares as.

24² = 7² – 5² = 5² – 1²

3² = 9² – 7² = 6² – 2²

40 = 11² – 9² = 7² – 3²

(i) Explain the method to write the multiples of 8 from 24 onwards in this method on the basis of algebra.

(ii) In how many methods the multiples of 16 from 48 onwards to write the difference of perfect squares?

Answer 1

(i) 4xy = (x + y)² – (x – y)²

24 = 4 × 6 × 1 = ( 6 + 1)² – (6 – 1)²

= 7² – 5²

24 = 4 × 3 × 2 = (3 + 2)² – (3 – 2)²

= 5² – 1²

The multiples of 8 from 24 onwards can be written in two forms as 4 × x × y

Number = 4 × y = (r + y)² – (r – y)²

Number = 4 ab = (a + b)² – (a – b)²

32 = 4 × 8 × 1

= (8 + 1)² – (8 – 1)²

= 9² – 7²

32 = 4 × 4 × 2

= (4 + 2)² – (4 – 2)²

= 6² – 2²

40 = 4 × 10 × 1

= (10 + 1)² – (10 – 1)² = 11² – 9²

40 = 4 × 5 × 2

= (5 + 2)² – (5 – 2 )² = 7² – 3²

(ii) 48 = 4 × 4 × 3, 4 × 12 × 1, 4 × 6 × 2

There are the different ways of writing 48.

So it can be written in 3 different methods as the difference of perfect squares.

48 = 4 × 4 × 3

= (4 + 3)² – (4 – 3)² = 7² – 1²

48 = 4 × 12 × 1

= (12 + 1)² – (12 – 1)² = 13² – 11²

48 = 4 × 6 × 2

= (6 + 2)² – (6 – 2)² = 8² – 4²