A nice number is an integer ending in 3 or 7 when written out in decimal. Prove that every nice number has a prime factor that is also a nice numbers. My teacher just asked me a question like this but i do not know how to start and work it out at all. Can someone help me out with that?

Hint: The number is odd, so all of its prime factors are odd. Could they be all nasty (end in $1$ or $5$ or $9$? Examine products of nasty numbers. You will find they are all nasty. 
Remark: Note that a nice number can have some nasty prime factors. For example, $77$ has the nasty prime factor $11$, but it also has the nice prime factor $7$. Note also that the product of nice numbers can be nasty, example $3\cdot 7=21$. 
