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, $x_1.x_2.x_3.x_4=770=\mathrm{2.5.7.11}$
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 $(1,0,0,0),(0,1,0,0)(0,0,1,0)$ and $(0,0,0,1),$ similarly for other powers.
Hence total number of ways $=4\times 4\times 4\times 4=256$