Question

For how many natural numbers a is the expression 
$\sqrt{\frac{a+64}{a-64} }a−64a+64$
​​ also a natural number?​

Answer 1

Answer:

Thus, n2−1 is an odd number dividing 64, i.e. n2−1=±1. The only solution there is n=0,a=−64 - but this is not a natural number.

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Answer 2

Answer:

Answer: Only possible value of a is 1 .

Step-by-step explanation:

\sqrt{a + 64 / a- 64 }

a+64/a−64

By Bodmas rule , we have :-

=> \sqrt{a + \frac{64}{a} - 64 }

a+

a

64

−64

=> \sqrt{\frac{a^2 + 64 - 64a}{a} }

a

a

2

+64−64a

Since the expression is a rational number , then the expression \frac{a^2+64 - 64a}{a}

a

a

2

+64−64a

must be a perfect square .

Suppose \frac{a^2 + 64 - 64a}{a} = k^2

a

a

2

+64−64a

=k

2

for some k .

Then a² - 64a + 64 = ak²

Notice that by using a bit of logic , the RHS side is divisible by a , so the LHS side must also be divisible by a .