7. A natural number $N$ ends in ...70, and it has exactly 72 natural divisors (including 1 and itself). How many natural divisors does the number 80N have? ANSWER: 324.
Solution: Each divisor of the number $\mathrm{N}$ can be represented in the form $2^{a} \cdot 5^{b} \cdot q$, where q is not divisible by 2 and 5, and the numbers $a, b$ are 0 or 1. Therefore, there are only 4 different combinations for the pair a and b, which means there are $72 / 4=18$ possible values for q. These correspond to divisors of the number $80 \mathrm{~N}$ in the form $2^{c} \cdot 5^{d} \cdot q$, where c=0,...,5 and $d=0,1,2$. There are 18 for each q, i.e., a total of $18 * 18=324$.