1. Prime factorization of the product

We need 5-digit numbers whose digits have product $36$.

$36 = 2^2\cdot 3^2$

Since the product is nonzero, no digit can be $0$. Also, every digit must be a single-digit factor of $36$, so possible digits are from

$\{1,2,3,4,6,9\}$

because these are the one-digit numbers whose prime factors involve only $2$ and $3$ and divide $36$.

We need 5 digits whose product is $36$.

2. Convert the problem into factor partitions

We want 5 digits $d_1,d_2,d_3,d_4,d_5$ such that

$d_1d_2d_3d_4d_5=36$

Think of writing $36$ as a product of 5 one-digit positive integers. Since multiplying by $1$ does not change the product, some digits can be $1$.

So we list all multisets of 5 digits from $\{1,2,3,4,6,9\}$ with product $36$.

3. Find all possible non-1 factor combinations

First, factor $36$ into factors bigger than $1$:

• $36=36$ (not allowed, not a digit)
• $36=6\cdot 6$
• $36=4\cdot 9$
• $36=2\cdot 18$ (18 not allowed)
• $36=3\cdot 12$ (12 not allowed)
• $36=2\cdot 2\cdot 9$
• $36=2\cdot 3\cdot 6$
• $36=3\cdot 3\cdot 4$
• $36=2\cdot 2\cdot 3\cdot 3$

Now pad with $1$'s so that total number of digits is 5.

Thus the valid multisets are:

1. $\{1,1,1,6,6\}$
2. $\{1,1,1,4,9\}$
3. $\{1,1,2,2,9\}$
4. $\{1,1,2,3,6\}$
5. $\{1,1,3,3,4\}$
6. $\{1,2,2,3,3\}$

These are all possibilities.

4. Count permutations of each multiset

Now count distinct 5-digit numbers formed by each multiset.

Case 1: $\{1,1,1,6,6\}$

Number of permutations:

$\frac{5!}{3!2!}=10$

Case 2: $\{1,1,1,4,9\}$

Number of permutations:

$\frac{5!}{3!}=20$

Case 3: $\{1,1,2,2,9\}$

Number of permutations:

$\frac{5!}{2!2!}=30$

Case 4: $\{1,1,2,3,6\}$

Number of permutations:

$\frac{5!}{2!}=60$

Case 5: $\{1,1,3,3,4\}$

Number of permutations:

$\frac{5!}{2!2!}=30$

Case 6: $\{1,2,2,3,3\}$

Number of permutations:

$\frac{5!}{2!2!}=30$

5. Add all cases

$10+20+30+60+30+30=180$

6. Final answer

The number of 5-digit natural numbers whose digits have product $36$ is

$\boxed{180}$

7. Comparison with stored answer

Stored correct answer = $180$.

Our derived answer is also $180$, so they agree.