Arithmetic Review: Integers (Part 3)
Alright let’s jump straight ahead and cover the final operator division.
When you take an integer a and divided by an integer b, where b is a divisor also known as a factor
of a, the result is always a divisor of a. For example when 60 is divided by 6 (one of
its divisors), the result is 10 which is another divisor or factor of 60. Now where it gets
interesting is when b is not a divisor of a in this case the result can be viewed in
three different ways. It can be viewed as a fraction, as a decimal, or the result can
be viewed as a quotient with a remainder, where both the quotient and remainder are
integers. Fractions and decimals are useful when the
result must be viewed as a single number, while quotients with remainders are useful
for describing the result in terms of integers only. Since we are reviewing integers in this
video its natural to talk about them first. We will discuss fractions and decimals in
a later video. For now let’s discuss the third view, division as a quotient with a
remainder. Say I have the integer 13 and I want to divide
it by 5 notice that it won’t divide evenly with 5, you can also think about this in the following
way pretend you have 13 dollars and you are trying to divide this amount among 5 people
at least each person will receive 2 dollars, and you will have 3 dollars remaining. We
can express the number 13 by using the quotient in this case 2 and its remainder as follows:
13 = 2 times 5 + 3 hence the result of 13 divided by 5 is the quotient 2
or simply as “2 remainder 3”. In general when a positive integer say a is
divided by a positive integer b, you first find the greatest multiple of b that is less
than or equal to a. This particular multiple can be expressed as qb, where q is the quotient,
and the reminder is equal to a minus that multiple of b, notice that the remainder r
is always greater than or equal to 0 and less than b.
For example 100 divided by 45 is 2 remainder 10, since the greatest multiple of 45 that’s
less than or equal to 100 is 2 times 45, or 90 which is 10 less than 100.
In the next example,
24 divided by 4 is 6 remainder 0 since the greatest multiple of 4 that’s less than
or equal to 24 is 24 itself, which is 0 less than 24. The fact that we got a remainder
of 0 means that 4 is a divisor or factor of 24.
How about the following division, 6 divided by 24 is 0 remainder 6, since the greatest
multiple of 24 that’s less than or equal to 6 is zero times 24, or 0 which is 6 less
than 6. These are the 3 distinct cases when dealing
with divisions involving quotients and reminders. Lets do one more of each type just to make
sure you understand this: Lets try 100 divided by 3 this will be equal
to 33 remainder 1 since 100 divided by 3 goes evenly 33 times so we take 33 multiply it
by 3 then we add 1 to the product to write it as: 100 =(33)(3)+1
Lets try the second case: 100 divided by 25 is 4 remainder 0, since
100 divided by 4 goes evenly 4 times so we take 4 and multiply it by 25 and obtain 100
so there is no need to add a remainder, in this particular case the remainder is zero.
So 100 is written as:
(4)(25)+0 And finally the third case: 100 divided by
150 is 0 remainder 100, since 100 divided by 150 goes evenly 0 times so zero times 150
is equal to 0 so we need to add 100 which itself represents the remainder,
so 100 is written as: (0)(150)+100
Alright, now Let’s talk about divisibility of integers:
Here are some rules to help you determine if a number is divisible by a certain integer:
An integer is divisible by 2 if its units digit is divisible by two. For example, we
know just by glancing at it that 1,562,158 is divisible by 2, because the units digit
8 is divisible by 2. Likewise the integer 786 is also divisible by 2 because the unit
digit 6 is divisible by 2. An integer is divisible by 3 if the sum of
its digits is divisible by 3. For example we know that 5616 is divisible by 3 because
the sum of its digits 5+6+1+6 = 18 which is divisible by 3. Also we can add the digits
of 18 as 1 + 8 which is equal to 9 which is also divisible by 3.
An integer is divisible by 4 if its last two digits form a number that’s divisible by
4. For example, 712 is divisible by 4 because the last two digits 12 is divisible by 4.
Likewise the integer 358916 is divisible by 4 because the last two digits 16 are divisible
by 4. An integer is divisible by 5 if its units
digit is either 0, or 5 like in 105 or 20,180. An integer is divisible by 6 if its divisible
by both 2, and 3 like in 204, this integer is divisible by 2 because its unit digit is 4, and is divisible
by 3 because the sum of its digits 2 + 0 + 4 = 6 which is divisible by 3 hence the
integer 204 is divisible by 6. An integer is divisible by 9 if the sum of
its digits is divisible by 9. Like in the number 61326 if we add the digits 6+1+3+2
+6 we obtain 18 which is divisible by 9, and finally an integer is divisible by 10 if its
units digit is 0. Like in the number 100 or 145890.
Rules for divisibility by 7 and 8 do exist but are too complicated and not worth memorizing.
Nevertheless they do exist. Alright in our next video we will focus on
the concepts and terminology that are associated with special integers.