What is the least positive integer by which $2^5\times 3^6\times 4^3\times 5^3\times 6^7$ should be multiplied so that the product is a perfect square?

Let the number be $N=2^5\times 3^6\times 4^3\times 5^3\times 6^7$.
We find the prime factorization of $N$ by breaking down composite bases $4$ and $6$:

Substitute these back into the expression for $N$:

Combine the exponents for each prime base:

For a number to be a perfect square, all exponents in its prime factorization must be even integers.

• Exponent of $2$ is $18$ (even, needs no factors).

• Exponent of $3$ is $13$ (odd, needs to be multiplied by $3^1$).

• Exponent of $5$ is $3$ (odd, needs to be multiplied by $5^1$).

Thus, the least positive integer by which $N$ must be multiplied is: