Proof of 'No natural number whose multiplication of digits is equal to 3570' [closed]

Solution 1:

Suppose there were a natural number that satisfies this. Any natural number has the base 10 digit form $x_{1}x_{2}...x_{n}$ where we understand each $x_{j}$ to be a digit (not a product!). In particular $1 \leq x_{1} \leq 9$ and is an integer, and for $2 \leq j \leq n$, $0 \leq x_{j} \leq 9$ and is also an integer. Then we would have $$ \prod_{j=1}^{n}x_{j}=3570=2 \times 3 \times 5 \times 7 \times 17. $$

17 is a prime and hence on the left hand side of the above one would need an $x_{j}$ to be a multiple of 17 for the equation to hold. But every multiple of 17 is two digits or more which contradicts the definition of our $x_{j}$'s. Hence we have the result.