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

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