Of the three figures shown below, which ones can be drawn without going over any lines twice and without crossing any lines? 

We have:

\(\overline{a_na_{n-1}...a_3a_2a_1}=\overline{a_na_{n-1}..a_4}.1000+\overline{a_3a_2a_1}\)

Since \(1000⋮4\) => \(\overline{a_na_{n-1}..a_4}.1000\) \(⋮\) \(4\).

\(\overline{a_na_{n-1}...a_3a_2a_1}\)  \(⋮\) \(4\)  iff   \(\overline{a_3a_2a_1}\) \(⋮\)  \(4\)

In other word: a number is divisibility by 4 if only if the number formed by its three right digits is divisibility by 4.

Factors of N: {pⁱ q^j | i = 0, .., m ; j = 0, .., n}

=> Total: (m + 1)(n + 1) factors.

Factors of N:  {1, p, p², ...., p^k-1, p^k }

=> There are k + 1 factors.

five 5-student committees and one 7-student committees.

Total: 6 committees.

Each year has 365 or 366 days, so the least number of students should be 367 to be sure that there are at leat two students with the same birthday.