a decimal point and some digits. Round this up to 20. The primes less than 20 are 2, 3,
5, 7, 11, 13, 17, and 19. As you divide 373 by each of these primes, you’ll see that none
of them gives you a quotient without a remainder. That means 373 is itself a prime
number, so the ratio 373:273 is in lowest terms.
- You can use the “brute-force” method and factor both the numerator and the
denominator of 231/230 into primes, and you’ll discover that the prime factors that
make up the numerator are entirely different from the prime factors that make up the
denominator. Therefore, the fraction 231/230 has to be in lowest terms. But there’s
an easier way to see this, and you don’t have to do any work to figure it out. Note that
the absolute values of the numerator and the denominator differ by only 1, and the
denominator is positive. Now think: what will happen if you divide both the numerator
and the denominator by any positive integer other than 1, in an attempt to reduce the
fraction? The resulting numerator and denominator will always have absolute values
that differ by less than 1, so they can’t both be integers. But in order to be a “legitimate
fraction,” both the numerator and the denominator must be integers.
- You want to see if −154/165 is in lowest terms, and if it is not, to reduce it. First,
convert both the numerator and the denominator into products of primes and then
attach the extra “factor” of −1 to the numerator, like this:
− 154 =− 1 × 2 × 7 × 11
and
165 = 3 × 5 × 11
Next, use these products to build a fraction in which both the numerator and the
denominator consist of prime factors, and the numerator has the extra “factor” −1:
(− 1 × 2 × 7 × 11) / (3 × 5 × 11)
The common prime factor is 11. Remove it from both the numerator and the denominator,
getting
(− 1 × 2 × 7) / (3 × 5)
That’s −14/15. You can sense immediately that this is in lowest terms because the
numerator and the denominator have absolute values that differ by only 1, and the
denominator is positive.
- When you have two fractions in lowest terms and multiply them, the product is
sometimes in lowest terms, but not always. First, consider this:
3/5× 7/11 = (3 × 7) / (5 × 11)
= 21/55
This product is in lowest terms. You know this because the numerator is the product
of the primes 3 and 7, and the denominator is the product of the primes 5 and 11.
When the numerator and denominator of a fraction are both factored into primes, the
Chapter 6 603
