Question

The number of non-zero integral solution to ${{\left| 3+4i \right|}^{n}}={{25}^{n}}$ is
(a)1
(b)2
(c)Finitely many solution
(d)None of these

Answer 1

Hint: An Integral solution is a solution such that all the unknown variables take only integer values. Given three integers a, b, c representing a linear equation of the form ax + by = c. Determine if the equation has a solution such that x and y are both integral values.

Complete step-by-step answer:
The integers are the numbers {…, -3, -2, -1, 0, 1, 2, 3,}. There are infinitely many of them. The equation 3x = 2 has only the solution $x=\dfrac{2}{3}$ , which is NOT an integer, so it has no integer solutions. Integer solutions to an equation mean solutions which are in the set of integers.
Let us consider the given expression,
${{\left| 3+4i \right|}^{n}}={{25}^{n}}$
We know that, $\left| a+ib \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$
${{\left( \sqrt{{{3}^{2}}+{{4}^{2}}} \right)}^{n}}={{25}^{n}}$
\[{{\left( \sqrt{9+16} \right)}^{n}}={{25}^{n}}\]
\[{{\left( \sqrt{25} \right)}^{n}}={{25}^{n}}\]
The square root of the 25 is 5, we get
\[{{\left( 5 \right)}^{n}}={{25}^{n}}\]
\[{{\left( 5 \right)}^{n}}={{\left( {{5}^{2}} \right)}^{n}}\]
By using indices formula, we get
\[{{\left( 5 \right)}^{n}}={{\left( 5 \right)}^{2n}}\]
If the base is the same, then the powers are equal.
$n=2n$
$2n-n=0$
$n=0$
Thus, 0 is the only integral solution of the given equation. Therefore the number of non-zero integral solutions of the given equation is zero.
Hence, the correct option for the given question is option (d).

Note: We might get confused between positive integral solution and non-zero integral solution. A positive integer is any integer that is greater than zero. In other words, a positive integer is any integer that is to the right of zero on the number line. Non-zero integral solution states that the solution is not equal to zero and is strictly an integer.