Number theory is one of the oldest mathematical fields. Euclid published one of the oldest known algorithms. Take two rods of different lengths. Choose a unit, such that an integer number of them fit exactly. Which is the maximum number of those units, common to both? That is, this so called greatest common divisor (gcd), is the smallest 'new unit', which covers them exactly.
In some cases (relative primes), the 'new unit', is the same as the old unit, but not always.
To this day there are open Number Theory problems.
Visit Terry Tao's page, for new results by this UCLA math professor.
Computers love integers, one can learn to talk to them by using these numbers. This can be the beginning of a love affair between humans and machines.