Arithmetic Review: Integers (Part 2) In the previous video we talked about the
basics of integers and how to add and subtract integers.
Now let’s talk about multiplication. Recall that if two numbers have the same sign, there
product is positive. For example 5 times 6 = 30 and -5 times -6 also equals 30.
If two numbers have opposite signs, their product is negative, for example -5 times
6 = -30 and 5 times -6 also equals -30 In a nut shell, a positive times a positive
equals a positive, a negative times a negative equals a positive, and a positive times a
negative equals a negative. That’s all there is to know about multiplication,
Now let’s talk about the product of integers them self’s, when integers are multiplied,
each of the multiplied integers is called a factor also known as a divisor of the resulting
product. For example: 6 times 5 = 30 this means that 6 and 5 are factors or divisors
of 30. Likewise in the product (2)(3)(10) = 60, the integers 2, 3 and 10 are factors
or divisors of 60. But notice that 4, 15, 5 and 12 are also factors of 60 since (4)(15)
= 60 and (5)(12) = 60. As the result the positive factors of 60 are 1,2,3,4,5,6,10,12,15,20,30,
and 60. Keep in mind that the negative of these integers are also factors of 60 since
(-2)(-30) and (-4)(-15) also results in 60. Now let’s focus on multiples.
A multiple of an integer is that number multiplied by an integer for example: the integers -20,-10,10,20,30,40,50, and 60
are all multiples of 10, since the previous set of numbers can be obtain by taking 10
and multiplying it by an integer. Also keep in mind that, technically the number zero
is a multiple of every number. For example 0 times 10 will give you 0 and 0 times -10
also gives you zero. In general a multiple is divisible by each of its divisors without
leaving a remainder. Also keep in mind that there are only a few
factors of any integer and there are infinitely many multiples of any integer.
For example to find the factors of 100 we start with 1 times 100 and systematically
write down the factors in pairs whose product equal 100, the next pair is 2 times 50, then
4 times 25, followed by 5 times 20, and finally 10 times 10, so the factors of 100 are, 1,2,4,5,10,20,25,50, and 100
and there opposites -1,-2,-4,-5,-10,-20,-25,-50 and -100. As a consequence 100 is a multiple
of 18 integers 9 positive and 9 negative. The multiples of 100 includes positive numbers
such as 100,200,300,400, and so on, along with its negative multiples such as -100,-200,-300,-400
and don’t forget 0 which is technically a multiple of every number. So every non zero
integer has infinitely many multiples. Some final detail about factors and multiples
to keep in mind is that 1 is a factor of every integer; with that said 1 is not a multiple
of any integer except 1 and negative 1. And remember 0 is a multiple of every integer;
and 0 is not a factor of any integer except 0.
Now that we know about factors and multiples let’s talk about the least common multiple
of two nonzero integers a and b. The (LCM) of these two integers
is the least positive integer that is a multiple of both a and b. For example, the least common
multiple of 30 and 75 is 150. One strategy to find the LCM is to find and list Positive
multiples of 30 and 75 as follows: the multiples of 30 are 30, 60, 90, 120, 150, 180, 210,
240, 270, 300, … and so on and the multiples of 75 are 75, 150, 225, 300, 375, 450 and
so on, thus, the common positive multiples of 30 and 75 are 150, 300, 450, and so on
and the least of these multiples is 150 hence the LCM of 30 and 75 is equal to 150.
Like wise to find the LCM of 40 and 100, we list the first couple multiples of 40 and 100,
as follows the multiples of 40 are 40,80,120,160,200,240,.. and so on, the multiples of 100, are 100,
200, 300, 400, and so on the least that is common to both 40 and 100 is 200 hence the
LCM of 40, and 100 is 200. Now lots talk about the GCF,
The greatest common divisor also known as the greatest common factor of two nonzero
integers a and b is the greatest positive integer that is a divisor also known as a
factor of both a and b. For example: the positive common divisors or factors of 30 are: 1,2,3,5,6,10,15
and 30, and for 75 they are: 1,3,5,15,25,75. Thus the common positive divisor or factor
of 30 and 75 are 1,3,5 and 15 so we choose the greatest of these which is equal to 15.
Like wise the GCF of 40 and 100 can be found by listing all the factors of 40, which are
1,2,4,5,8,10,20, and 40, and list the factors of 100, which are: 1,2,4,5,10,20,25,50,100
thus the common positive divisors or factors are 1,2,4,5,10, and 20 so we choose the greatest
of these factors which is equal to 20. Alright in our next video we will focus on
the concepts and terminology that are associated with 4th basic operator known as division.