Notes for June 16

Here are the topics we covered Monday.

Primes, composites and the unit (1)

How many factors does a number have.

1 has one factor, just itself. (Every number has itself as a factor.)
2 has two factors, 1 and 2.
3 has two factors, 1 and 3.
4 has three factors, 1, 2 and 4.

This gives us examples of the three possible cases.

A unit has exactly one factor. The only unit is the number 1.

A prime has exactly two factors, itself and 1.

Any number with more than two factors is called a composite.


Follow this link to see previous posts about Prime factorization.

If we want to check to see if a number k is prime, we have to see if any prime number less than the square root of k divides evenly into k.

Two examples.

Is 87 prime? We have to check all primes less than sqrt(87) =~ 9.327..., so that means checking 2, 3, 5 and 7.

87/2 = 43.5, so 2 doesn't work.

87/3 = 29. Now we can stop, because 87 is a composite 3 x 29.

Is 89 prime? We have to check all primes less than sqrt(89) =~ 9.44398..., so that means checking 2, 3, 5 and 7.






89/2 = 44.5, so 2 doesn't work.

89/3 = 29.666..., so 3 doesn't work.

89/5 = 17.8, so 5 doesn't work.

89/7 = 12.714..., so 7 doesn't work.

We can stop, 89 is prime.

We also were introduced to the concept of relatively prime. Numbers a and b are relatively prime if they have no primes in common in their respective prime factorizations.

If we have a fraction, we can reduce it to lowest term a/b, where a and b are relatively prime.



Follow this link for more information on Relatively prime pairs and reducing fractions to lowest terms .

Representing numbers by prime factorization

I'm going to be putting previews of notes for class before the class, then fleshing in the details and possibly adding more or taking away topics from the list, depending on how far we get.


Prime numbers, composite numbers and the number 1 (unit)
prime factorization
relatively prime numbers
shortcuts for divisibility
reducing fractions
greatest common divisor (g.c.d.) and least common multiple (l.c.m.) for two numbers

These topics are covered on pages 23 through 28 in the notes.