Question

If $N$ is the number of positive integral solutions of the equation $x_{1}x_{2}x_{3}x_{4}=770$ , then the value of $N$ is

• A. $250$
• B. $252$
• C. $254$
• D. $256$

Answer 1

Answer: (D)

Given, $x_{1} . \, x_{2} . \, x_{3} . \, x_{4} = 770 = 2.5.7.11$
Let positive integer $x_{1} = 2^{a_{1}} .5^{b_{1}} . 7^{c_{1}} .11^{d_{1}}$
Let positive integer $x_{2} = 2^{a_{2}} .5^{b_{2}} .7^{c_{2}} .11^{d_{2}}$
Let positive integer $x_{3} = 2^{a_{3}} .5^{b_{3}} .7^{c_{3}} .11^{d_{3}}$
Let positive integer $x_{4} = 2^{a_{4}} .5^{b_{4}} .7^{c_{4}} .11^{d_{4}}$
Now, $x_{1} x_{2} x_{3} x_{4} = 2^{a_{1} + a_{2} + a_{3} + a_{4}} .5^{b_{1} + b_{2} + b_{3} + b_{4}} .7^{c_{1} + c_{2} + c_{3} + c_{4}} .11^{d_{1} + d_{2} + d_{3} + d_{4}}$
As per given condition $a_{1} + a_{2} + a_{3} + a_{4} = 1 ,$ which can be in $4$ ways $\left(\right. 1 , \, 0 , \, 0 , \, 0 \left.\right) , \, \left(\right. 0 , \, 1 , \, 0 , \, 0 \left.\right) \left(\right. 0 , \, 0 , \, 1 , \, 0 \left.\right)$ and $\left(\right. 0 , \, 0 , \, 0 , \, 1 \left.\right) ,$ similarly for other powers.
Hence total number of ways $=4\times 4\times 4\times 4=256$