Question

If $N$ is the number of positive integral solution of $x_1{x}_2{x}_3{x}_4=770$

• A. $N$ is divisible by $4$ distinct primes
• B. $N$ is a perfect square
• C. $N$ is a perfect $4th$ power
• D. $N$ is a perfect $8th$ power

Answer 1

Answer: (B), (C), (D)

$N$ is a perfect square
$N$ is a perfect $4th$ power
$N$ is a perfect $8th$ power

Answer is B C D.

We have $770=2\times 5\times 7\times 11$

We can assign $2$ to $x_1$ or $x_2$ or $x_3$ or $x_4$. Thus $2$ can be assigned in $4$ ways. Similarly each of $5,7,11$ can be assigned in $4$ ways.

Thus the number of positive integral solution of $x_1{x}_2{x}_3{x}_4=770$ is ${=4}^4{=2}^8=256$