ding-dong Michael here I hope that you
are all practicing responsible social
distancing right now it is so important
that we all do our part in times like
these we need to be unified but today
we're going to divide if you want to
know if a certain number can be divided
by another number with no remainder left
over well you could use a calculator or
you could ask Google but where's the fun
in that look today we are going to take
a stroll through some divisibility rules
for the numbers 1 through 9 these
procedures are a lot of fun they allow
us to quite quickly determine whether a
given number can be evenly divided by
another number that is with no remainder
left over now a calculator is sometimes
faster than these procedures I'll admit
that
but these procedures allow us to do two
things they allow us to develop our
number sense and they also allow us in
these lonely times to make a bunch of
new friends math friends let's get
started now it's pretty easy to check if
a number is divisible by one two or five
because if the number is an integer if
it has a no decimal or fractional
component well then it can be divided
into groups of one if it is even it's
divisible by two and if it ends in a
zero or five its well it's divisible by
five now a lot of us also know a trick
for checking divisibility by 3 the trick
is to take all the digits in the given
number and just sum them up if their sum
is divisible by 3 so is the actual
original number let's take a look at
this in action first let's pick a number
completely at random like oh I don't
know how about 360 2880 is this number
divisible by 3 can I take this many
things group them all into threes and
have nothing left over well let's apply
our trick we just simply sum up the
digits in the number we have 3 plus 6
which is 9 plus 2 which is 11 plus 8
which is 19 plus 8 which is 27 plus 0
which keeps us at 27 now because 27 is
divisible by three it happens to be
three times nine so is this original
number so that helps us take care of
divisibility by three
what about divisibility by nine well as
it turns out this trick works there too
if the sum of every digit in the
spelling of the number sums up to a
number that is divisible by nine
well then the original number is as well
but what about six well for six all you
need to do is make sure that it is
evenly divisible by three and that it's
even so in this case we know that three
hundred sixty-two thousand eight hundred
and eighty is divisible by 3 because
this trick shows us that but it is also
even therefore it is divisible by six as
well okay now I think it is time for us
to move on to divisibility by four we'll
keep using the same number three hundred
and sixty two thousand eight hundred and
eighty can I divide this into groups of
four with no remainder well here's how
you can check if the last two digits of
a number form a number that is divisible
by four then the entire number is eighty
is easily divisible by four right that's
just 4 times 20 now let's say though
that we had a different number like
three hundred and sixty-two thousand
eight hundred and ninety six let's say I
don't know right away whether 96 is
divisible by four well here's a little
bit of a trick you can use to figure out
if it is simply take the tens digit of
this two-digit number and double it
which means two times nine and then add
to that the ones digit six if their sum
is divisible by four then the original
number is as well in this case we have
two times nine which is 18 plus six
this gives us 24 and 24 is divisible by
four so both our original number and
this new one can be divided into groups
of four with no remainder pretty cool
what about eight well if I want to know
if a number can be divided evenly by
eight I need to look not at the last two
digits but at the last three here's what
I do I take the hundreds digit and I'm
old
apply it by four so this is four times
eight to that I add the tens digit times
to the tens digit is eight and then to
that I add the ones digit zero 4 times 8
is 32 2 times 8 is 16 and 0 is just 0
now 32 plus 16 is 48 is 48 divisible by
8 yes it is it happens to be equal to 8
times 6 so our original number is evenly
divisible by 8 okay so we have covered
divisibility by 1 2 3 we've done 4 & 5 &
6 & 8 & 9
whoa-ho-ho what about 7 well 7 is very
very fun here's what you do with 7 I
think that for a number as long as our
starting number which has 6 digits we're
going to probably have to do this
procedure multiple times but all of
these tricks can just be done repeatedly
you just do it once to the original
number and then if you still don't quite
recognize what you're left with and
whether it's divisible by the target
number or not guess what just keep
applying the trick until you see
something you recognize we're gonna
definitely have to do that when we check
divisibility by 7 for 350 2880
but here's all you have to do take a
look at the number and look at the ones
digit
now take that ones digit and multiply it
by 5 okay and then add to this what's
left as if it's a number all by itself
which means we're going to add 5 times 0
2 3 6 2 8 8 all right now we sum these
up and if their sum is divisible by 7
then the original number is as well so
what do we got here well we've got this
number plus 5 times zero which is zero
so it's just the same number now I'll be
honest I'll admit this I'm not afraid to
I don't know if this is divisible by 7
or not actually I do because it's not a
coincidence that literally every single
one of these digits evenly goes into
this number 360 2880 happens to be
nine factorial which means it is equal
to one times two times three times four
times five times six times seven times
eight times nine therefore it is a
multiple of all of them but let's show
that using this trick I can apply the
trick again to the result here all that
means I need to do is I take the ones
digit I multiply it by five so that's
five times eight and then I add to this
expression what's left which is just
three six to eight now five times eight
is equal to 40 and this number is three
thousand six hundred and twenty-eight if
I sum them together what I get is three
thousand six hundred and sixty-eight
alright if I still don't know if this is
divisible by seven I just keep going so
we take the ones digit alright and we
multiply it by five eight times five
again as we know is 40 and then I add
what's left three hundred and sixty six
okay this gives us four hundred and six
is four hundred and six divisible by
seven if I still don't know we just keep
going
what is six times five well we'll deal
with that later we just know that we
need to take five times the ones digit
and summit with what's left 40 now 5
times 6 is 30 and 30 plus 40 is 70 70
II's aliy recognized as being divisible
by seven it is just seven times ten
therefore because this is divisible by
seven our original number is two pretty
cool very fun but how do these tricks
work oh well that's the fun part let's
get into it let's start with
divisibility by three why should it be
the case that summing all of the digits
used in the spelling of a number and
then checking if that sum is divisible
by three tells us anything about the
divisibility by three of the original
number well let's generalize this and
just imagine a number that has let's say
three digits this works for you know a
number
with any number of digits though but
let's say that this is our three digit
number ABC where a is the hundreds digit
B is the tens digit and C is the unit's
digit now what ABC means when we write
it down
we don't actually mean yet ABC that
means you know a number of things equal
to a plus B plus C no no instead we use
a base-10 positional system where this
actually means 100 times a plus 10 times
B plus C so if this number is say 417
well then we're going to have four 100's
110 and seven units but let me ask you
this what would it take for this the sum
of the digits in the number to equal the
actual amount the number represents well
we'd have to add more stuff to this we
only have one a here but we need 100 so
we're going to need 99 Moraes now we
only have one B but we need 10 so that
means we're gonna need nine more B's we
already have a seat so we don't need to
add any more C's there you go this right
here is equal to this the actual amount
represented by the original number oh
look at this
this is very interesting 99 a plus 9b
this part of the expression will always
be evenly divisible by three no matter
how many digits are in the number that
you are checking every coefficient in
this part will just be a string of nines
which means each term in it will be
divisible by three and if you add a
bunch of groups of three to another
bunch of groups of three well what
you're left with is just a bunch of
groups of three what you're left with is
a sum that is still divisible by three
and so because this component is always
divisible by three
all we need to do is check whether this
part is if it is well then the original
number is the sum of two numbers
divisible by three so it itself will be
visible by three in a similar way the
divisibility check for four operates as
well so let's check that let's again say
that we have a number like a and B and C
now remember that when we check for
divisibility by four we only concerned
ourselves with the last two digits why
would that be well it is because
conveniently four goes into 100 an
integer number of times 25 in fact so
any number that ends with at least two
zeros is divisible by four for example
the number let's say one seven nine two
zero zero I just made this one up but I
know that four will evenly go into this
because it is just 1792 hundreds and
four goes into a hundred what that means
is that if we when we take a number like
the number that we started with 360 2880
it doesn't end with two zeros but we can
think of it as being three hundred and
sixty-two thousand eight hundred which
we know that 4 goes into plus 80 so we
only need to check the divisibility of
this component those last two digits 80
all right so remember we took BC right
we took the the last two digits and we
doubled the tens digit and then we added
the ones digit why does this work well
in the same way that the check for
divisibility by three works let's
remember that BC we're forgetting about
a for now BC means 10 times b plus c
number of things well how do we get from
this checking expression to the actual
value of BC well we're going to need
eight more B's and no C's we'll look at
that
8 8 b 8b will always be divisible by 4
because 8 can be put into groups of 4
and we're gonna have b times as many of
those groups of four so we only need to
know if this other part is divisible by
4 and if it is well then the end
higher number is you might think oh why
don't we just use this trick to check
for divisibility by eight I wish that we
could but the problem is that this
entire trick relies on the fact that the
last two digits of the number are all
that matter because four goes into 100
but eight does not however eight does go
into 1000 eight times 125 equals 1000 so
if a number ends with three zeroes or
more we know that it is evenly divisible
by 8 we'll need to worry then about the
last three digits of any number to check
for divisibility by eight if you're
thinking about a number that doesn't
have three digits that it's only a
two-digit number
well then you know you you probably can
just figure out if it's divisible by 8
by simply knowing your 8 times tables if
you know them if you've memorized them
up to eight times ten at least well
that's already 80 from there you go from
80 to 88 and then you go to 96 and then
you're done those are the only other
two-digit multiples of eight 96 is eight
times 12 of course but when it comes to
a three-digit number I'll say ABC
doesn't matter what other digits are out
here if any we check divisibility by
eight by taking four times the 100's
digit and summing that with two times
the tens digit and then just the unit's
digit by itself why does this allow us
to check for divisibility by eight well
let's take a look we know that the
actual amount represented by a BC is
equal to 100 times a plus ten times B
plus C to get from this expression the
checking expression to the actual amount
represented by ABC we need 96 mores we
need eight more B's and no more C's what
we just talked about how 96 is 8 times
12 and of course 8 goes into 8 so this
part is always divisible by 8 regardless
of what a and B are so we only need to
make sure that this expression is
divisible by 8 if it is then the
original
number is okay you know what all we have
left to discuss now is divisibility by
seven why did that trick work it was
kind of strange right we took the ones
digit of the number multiplied it by
five and then added it to what was left
well the trick to kind of wrapping your
head around why this works is to figure
out what it means to sum what's left
after removing the ones digit so let's
say that we have a number like a b c
doesn't have to be three digits long
could do it of any length but we take
this ones digit c and we multiply it by
five and we add that to what's left how
do we represent this procedure of taking
what's left and treating it like it's
its own number well one way to do that
is to think about the original number a
b c and then subtract C from it okay
this would allow us to take a number
like 417 and then subtract that last
digit seven if we subtract seven then
we're left with 410 we've turned the
ones digit into a zero then we we want
to just get rid of it completely and we
can easily do that by dividing by ten in
which case we then wind up with 41 so
we're going to have to divide this by 10
which is the same as multiplying by 1/10
this looks a little bit messy and I hope
you're able to follow it all we're
trying to do is algebraically show what
is done when we ignore the ones digit
and treat what's left is its own number
that's what we're doing here we're
taking away the ones digit so it becomes
a zero and then dividing by 10 so we
turn ABC into just a be perfect now how
does this and whether it's divisible by
seven tell us that this is also the
original number is also divisible by
seven oh boy well this is very fun let's
play around with this expression and see
what else it is either equal to or
divisible by the same numbers as okay so
let's distribute the 1/10 first because
1/10 times the stuff in parentheses can
also be represented as
tenth time's ABC minus 1/10 times C all
right then we still have to add the five
see at the end oh look at this we've got
ourselves two different amounts of C we
can combine those what is five C minus
1/10 of a C well that's five minus 0.15
minus point one is a positive four point
nine we still have this 1/10 ABC so this
expression right here
this third one that we've got is equal
to this expression which is equal to
this expression that we actually are
using to check for divisibility by seven
you might already have a bit of a sense
of why this works because 4.9 looks an
awful lot like 49 a number that is
famously divisible by seven let's go
ahead and make this even easier by
taking this and multiplying it by ten
multiplying by ten doesn't change what
this can be evenly divided by it just
means that whatever amount of groups we
can make we're gonna have ten times more
of them right it doesn't add any
remainders so we're fine if we multiply
this whole thing by ten that means we're
multiplying 1/10 ABC by ten ten times
1/10 is one so we've got a B C and then
10 times four point nine well that's
just 49 C oh look at this
alright so this thing that we've wound
up with is divisible by the same numbers
as this which is divisible by the same
numbers as this which is divisible by
the same numbers as this the actual
procedure we use to check for
divisibility by seven but look at this
guy this is pretty clearly made up of a
part that is always going to be
divisible by seven 49 49 things can be
put into groups of seven and we're gonna
have C times more of those groups so if
this if this if this and if this happen
to be divisible by set
well then that means that ABC must be
too but ABC is the original number so if
this checks out and gives us a number
divisible by seven then the number we
started with was too beautiful I love
this kind of stuff if you've got some
time while you're quarantined I highly
recommend playing around with numbers
and coming up with some of your own
tests of divisibility it's very fun I
did it for six and I loved it obviously
this doesn't necessarily make math
faster to do a calculator will be faster
almost all the time except if you've if
you've memorized things but the point
isn't speed the point of these exercises
is to develop our number sense and come
closer to this thing we call number now
I'd like to end with a word from our
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with something that we were able to make
and include in it in 1933
Theodore Edison son of Thomas Edison
invented a puzzle that he called the
caliber on 12 and we went back to the
original took the measurements and
created a replica of the caliber on 12
look at that whoop very beautiful the
pieces are all of the same size as the
original and they're even made of the
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excited about this puzzle because it is
I mean totally serious it is the only
puzzle that I own that I have yet to
solve I have a wooden version of this
puzzle that's a little bit different
than the original and I've had it for a
couple of years and I still have not
figured it out the
puzzle goes like this you take these 12
sort of burgundy pieces and you need to
assemble them into a rectangle that was
the original puzzle but the puzzle came
in a box just like this and the pieces
all came flat in the box but this box's
shape is not the shape of the rectangle
that the 12 burgundy pieces fit into
this is a different dimension rectangle
and in order to make these twelve pieces
fit into here you need to use one of
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called spacers so all twelve pieces came
in the original box along with a spacer
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somehow all 12 of these burgundy pieces
and this 13th black spacer piece can be
fit into the box once you've solved that
puzzle replace this spacer with a
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try to fit them all in it's possible
when you're done with that puzzle then
fit this nice little skinny nude in
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the box again that's three different
puzzles but the original puzzle was just
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put them into a rectangle now what makes
me so excited is that this is actually
this actually comes with more puzzles
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bought the original back in the 30s from
Theodore Edison it only came with one
spacer and you didn't know which type of
spacer you were going to get we have
reproduced all three of the spacers he
made and so now you can feast your brain
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