Question

if the sum of all positive rational numbers n such that √n²+84n+1941 is an integer is 189/k‚then find k

Answer 1

Answer:

Step-by-step explanation:

Let the integer given by the square root be represented by = $m$

Then, $\sqrt{n^{2}+84n+1941}=m\\\\\Rightarrow{{n^{2}+84n+1941}=m^{2}}\\\\\Rightarrow{{n^{2}+84n+1941}-m^{2}=0}$................ equation 1

For Quadratic Equation $ax^{2}+bx+c=0 \text{ [where x is the variable and a, b and c are known values]}$

the value of $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

$b^{2}-4ac \text{ is called Discriminant (D)}$

In equation -1

${n^{2}+84n+1941}-m^{2}=0$

the value of $n=\frac{-84\pm\sqrt{84^{2}-4\times1\times(1941-m^{2})}}{2\times1}\\\\\Rightarrow{n}=\frac{-84\pm\sqrt{(2\times42)^{2}-4\times1\times(1941-m^{2})}}{2\times1}\\\\\Rightarrown=\frac{-84\pm\sqrt{4\times[{42^{2}-(1941-m^{2})]}}}{2\times1}\\\\\Rightarrown=\frac{-84\pm2\sqrt{[{42^{2}-(1941-m^{2})]}}}{2\times1}\\\\\Rightarrown={-42\pm\sqrt{42^{2}-(1941-m^{2})}$

$\Rightarrown={-42\pm\sqrt{1764-1941+m^{2})}={-42\pm\sqrt{m^{2}-177)}$

For the rational solutions for $n$, we rejected negative part of solution.

Then $n=-42+\sqrt{m^{2}-177)}$..........equation 2

As the n is a positive integer then the value of ${m^{2}-177)}$ is always positive.

Now consider

${m^{2}-177)=y^{2} \text{[where y is a positive integer]}$

${m^{2}-y^{2}=177\\\\\Rightarrow(m+y)(m-y)=177$...........equation -3

The factor pairs of 177 are (1 and 177) and (3 and 59)

Now we can rewrite the equation 3 as

$(m+y)(m-y)=177\times1$

Therefore $(m+y)=177$...........equation 4 and $(m-y)=1$ equation 5

Now

$(m+y)+(m-y)=1+177\\\\2m=178\\\\m=89$

$y=177-89=88$

Put the value of m in equation -2

$n=-42+\sqrt{m^{2}-177)}\\\\\Rightarrow{n}=-42+\sqrt{89^{2}-177}\\\\\Rightarrow{n}=-42+ \sqrt{7921-177}\\\\\Rightarrow{n}=-42+\sqrt{7744}\\\\\Rightarrow{n}=-42+88=46$

Again rewrite the equation 3 as

$(m+y)(m-y)=59\times3$

Therefore $(m+y)=59$...........equation 6 and $(m-y)=3$ equation 7

Now

$(m+y)+(m-y)=59+3\\\\2m=62\\\\m=31$

$y=59-31=28$

Put the value of m in equation -2

$n=-42+\sqrt{m^{2}-177)}\\\\\Rightarrow{n}=-42+\sqrt{31^{2}-177}\\\\\Rightarrow{n}=-42+ \sqrt{961-177}\\\\\Rightarrow{n}=-42+\sqrt{784}\\\\\Rightarrow{n}=-42+28=-14$[/tex]

As n is a positive integer then reject the value $n=-11$

and take the value $n=46$

As per the problem

$n=\frac{189}{k}\\\\\Rightarrow{k}=\frac{189}{n}=\frac{189}{46}$