If $$m$$ and $$n$$ are natural numbers such that $$n 1$$, and $$m^n = 2^{25} imes 3^{40}$$, then $$m - n$$ equals

Question

If $$m$$ and $$n$$ are natural numbers such that $$n > 1$$, and $$m^n = 2^{25} 	imes 3^{40}$$, then $$m - n$$ equals

Accepted Answer

We are given that $$ m $$ and $$ n $$ are natural numbers such that $$ n > 1 $$, and $$ m^n = 2^{25} 	imes 3^{40} $$. We need to find the value of $$ m - n $$.

#### Step 1: Express the right-hand side in terms of a single power
The goal is to rewrite $$ 2^{25} 	imes 3^{40} $$ in a form where all terms have the same exponent. Observe that the powers of $$ 2^{25} $$ and $$ 3^{40} $$ share a common factor, which is 5. This is because the greatest common divisor (GCD) of 25 and 40 is 5.

We can rewrite:
$$
2^{25} = (2^5)^5, \quad 3^{40} = (3^8)^5
$$

Thus, the expression becomes:
$$
m^n = (2^5)^5 	imes (3^8)^5 = \left(2^5 	imes 3^8ight)^5
$$

#### Step 2: Simplify the base
Calculate $$ 2^5 = 32 $$ and $$ 3^8 = 81 $$. Therefore:
$$
m^n = (32 	imes 81)^5
$$

Simplify further:
$$
32 	imes 81 = 209952
$$

Thus:
$$
m^n = (209952)^5
$$

#### Step 3: Determine values of $$ m $$ and $$ n $$
From the equation above, we can see that:
$$
m = 209952, \quad n = 5
$$

#### Step 4: Compute $$ m - n $$

Finally, subtract $$ n $$ from $$ m $$:
$$
m - n = 209952 - 5 = 209947
$$

### Final Answer:
Option C: **209947**

Answer

Answer

Answer