Answers for the problems at the end of Wednesday's class

Side of a triangle are 6, 11 and 8.

Scalene because all different.

6^2 + 8^2 = 100 < 11^2 = 121, so it is obtuse.

Area = 15sqrt(39)/4

Points are (2, 1) (5, 5) and (8, 9)

(2, 1) - (2, 1) = (0, 0)
(5, 5) - (2, 1) = (3, 4)
(8, 9) - (2, 1) = (6, 8)

Area =  ½| 3x8 - 6x4 | = 0

The three points are colinear.

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 .