Question

If BA × B3 = 57A, then the value of A and B is: (a) A = 5, B = 2 (b) A = 2, B = 5 (c) A = 5, B = 3 (d) A = 3, B = 5​

Answer 1

If BA × B3 = 57A, then the value of A and B is: option (a) A=5, B=2

Given,

BA * B3 = 57A

Solution,

From decimal number theory, we can write the value of BA as

$BA=(B*10^{1}+A*10^{0})\\BA=10B+A$

And the value of B3 would be,

$B3=(B*10^{1}+3*10^{0})\\B3=10B+3$

In a similar way,

$57A=(57*10^{1}+A*10^{0})\\57A=570+A$

Therefore, according to the question,

$BA * B3 =57A\\(10B+A) * (10B+3)=(570+A)\\100B^{2}+30B+10AB+3A=570+A\\100B^{2}+30B+10AB+2A=570\\-\frac{A}{5} =10B^{2}+3B+(A*B)-57$

Now, A=5 or A=0

If A=0

$10B^{2}+3B+(0*B)-57=0\\10B^{2}+3B-57=0\\B=\frac{-3+_-\sqrt{3^{2}-4*10*(-57)}}{2*10} \\B=\frac{-3+_-\sqrt{2289}}{20} \\B\neq integer$

If A=5,

$10B^{2}+3B+(5*B)-57=-1\\10B^{2}+8B-56=0\\5B^{2}+4B-28=0\\5B^{2}+4B=20+8\\5B^{2}+4B=5*2^{2}+4*2\\B=2$

Hence, A=5, B=2