Question

Question 4
When 5 is added to the natural number n it gives a perfect square and when 11 is subtracted from n it also gives a perfect square.
Find n.

Answer 1

Given : When 5 is added to the natural number n it gives a perfect square and when 11 is subtracted from n it also gives a perfect square.

To Find  :  n.

Solution:

5 is added to the natural number n it gives a perfect square

when 11 is subtracted from n it also gives a perfect square.

n + 5  = a²

n - 11    = b²

n + 5 - (n - 11) = a² - b²

=> 16  = a² - b²

=> 16 = (a + b)(a - b)

16  = 1  * 16   = 2 * 8  = 4 * 4

a + b = 16  a  - b = 1      => a = 8.5  not an integer

a + b = 8   a - b = 2   => a = 5   b = 3

n + 5  = 5²  = 25  => n = 20

n - 11    = 3² = 9  => n = 20

n = 20

a + b = 4   a - b = 4   => a = 4   b = 0

n + 5  = 4²  = 16 => n =  11

n - 11    = 0² = 0  => n =  11

n = 11

Hence n = 20  or  11

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