## Monday, January 7, 2013

### Relatively prime.

You might recall the idea of reducing fractions to lowest terms from school. I hope these memories are not accompanied by cold sweat or outbursts of obscenity. The name for the concept behind lowest terms is relatively prime.

Definition. Two numbers a and b are relatively prime to each other if their respective prime factorizations have no primes in common.

Let's use the idea in the concept where most people see it first, reducing fractions.

Example 1: 40% is the same as the fraction 40/100.  This is obviously not in lowest terms, since both are divisible by 10, so 40/100 = 4/10. This is still not in lowest terms since both are even. 4/10 = 2/5, and 2 and 5 are relatively prime. In fact, both 2 and 5 are prime, which might cause some confusion. Let's do a second example of relatively prime number where neither number is prime.

Example 2: 48% is the same as the fraction 48/100, not a relatively prime pair.  Let's reduce to lowest terms.

48/100 = 24/50 = 12/25, which is lowest terms.

12 = 3 × 2 × 2, so it isn't prime.
25 = 5 × 5, also not a prime.

The pair 12 and 25 is an example of a relatively prime pair of numbers where neither number is prime.

Tomorrow: The triangular numbers