Question

The sum of all natural numbers a such that a 2 16a + 67 is a perfect square is what?

Answer 1

Answer:

The sum of all natural numbers a such that a² - 16a  + 67 is a perfect square is  16

Step-by-step explanation:

a² - 16a  + 67  being a  perfect square

a²  - 2 * 8 a + 8²  + 3

(a - 8)²  + 3

this can be a whole sqaure

if (a - 8)² = 1

then 1 + 3 = 4 = 2²

(a - 8)² = 1

=> a -8 = ±1

=> a =  9 or 7

Sum of 9 +7 = 16

The sum of all natural numbers a such that a² - 16a  + 67 is a perfect square is  16

Answer 2

Answer:

16

Step-by-step explanation:

Given : The sum of all natural numbers a such that a² - 16a + 67 is a perfect square.

Taking the equation,

a² - 16a + 67 = a² - ( 8*2 a) + 8² +3

   This takes the form of a² - 2ab + b² = (a+b)²

Hence, we take it as  ( a-8)² + 3

We assume,  ( a- 8 )² = 1

Then,

1+3 = 4 =  2²

(a - 8)² = 1

=> a -8 = ±1

=> a =  9 or 7

Sum of 9 +7 = 16

Therefore,

The sum of all natural numbers a such that a 2 16a + 67 is a perfect square is 16