Question

if both 112 and 33 are factors of the number a * 43 * 62 * 1311, what is the smallest possible value of 'a'?

Answer 1

If both $11^2$ and $3^3$ are factors of the number a * $4^3$ * $6^2$ * $13^{11}$.

We have to determine the smallest possible value of 'a'.

Consider a * $4^3$ * $6^2$ * $13^{11}$

= $a * (2^2)^3 * (2 \times 3)^2 * 13^{11}$

=$a * 2^8 * 13^{11}$

So, a * $4^3$ * $6^2$ * $13^{11}$ can be expressed in prime factors as $a * 2^8 * 13^{11}$.

Since, $11^2$ is a factor of the given number.  If we do not include 'a' in the given number then 11 will not be a prime factor.

But as $11^2$ is a factor of the number, so,$11^2$ should be in 'a'.

As,$3^3$ is a factor of the given number and if we do not include 'a', since the number has only,$3^2$  in it. Therefore 'a' has to contain 3 in it. Therefore, 'a' should be at least $11^2 \times 3 = 363$.

The smallest value of 'a' is 363.