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# Divisibility tests for 2, 3, 4, 5, 6, 9, 10

Video transcript

What we're going
to do in this video are some real quick tests to see
if these three random numbers are divisible by any
of these numbers here. And I'm not going to focus a
lot on the why of why they're divisible-- we'll do that in
other videos-- but really just to give you a sense
of how do you actually test to see if this is
divisible by 2 or 5 or 9 or 10. So let's get started. So to test whether any of
these are divisible by 2, you really just have to
look at the ones place and see if the ones
place is divisible by 2. And right over here,
8 is divisible by 2, so this thing is going
to be divisible by 2. 0 is considered to
be divisible by 2, so this is going to
be divisible by 2. Another way to think
about it is if you have an even number
over here-- and 0 is considered to be
an even number-- then you're going to
be divisible by 2. And over here, you do not have
a number that is divisible by 2. This is not an even
number, this 5, so this is not divisible by 2. So I won't write any 2 there. So we've gone through the 2s. Now, let's work through the 3s. So to figure out if
you're divisible by 3, you really just have to
add up all the digits and figure out if the
sum is divisible by 3. So let's do that. So if I do 2 plus 7 plus 9
plus 9 plus 5 plus 8 plus 8, what's this going
to be equal to? 2 plus 7 is 9. 9 plus 9 is 18, plus 9 is 27,
plus 5 is 32, plus 8 is 40, plus 8 is 48. And 48 is divisible by 3. But in case you're not sure--
so this is equal to 48-- in case you're not sure whether
it's divisible by 3, you can just add
these digits up again. So 4 plus 8 is equal to 12, and
12 clearly is divisible by 3. And if you're not
even sure there, you could add those
two digits up. 1 plus 2 is equal to 3, and
so this is divisible by 3. This right over here,
let's add up the digits. And we can do this one in
our head pretty easily. 5 plus 6 is 11. 11 plus 7 is 18. 18 plus 0 is 18. And if you want to add the 1
plus 8 on the 18, you get 9. So the digits add up to 9. So these add up to 9. Well, they add to 18, which
is clearly divisible by 3 and by 9, and these two
things will add to 9. So the important
thing to know is when you add up all the digits,
the sum is divisible by 3. So this is divisible by 3
as well, divisible by 3. And then finally, Let's
add up these digits. 1 plus 0 plus 0 plus 7 is 8,
plus 6 is 14, plus 5 is 19. So we summed up the digits. 19 is not divisible by 3. So this one, we're not going
to write a 3 right over there. It's not divisible by 3. Let's try 4. And to think about
4, you just have to look at the last
two digits and to see-- are the last two
digits divisible? Are the last two
digits divisible by 4? Immediately, you can look
at this one right over here, see it's an odd number. If it's not going to
be divisible by 2, it's definitely not going
to be divisible by 4. So this one's not divisible by
any of the first few numbers right over here. But let's think about one, 88. Is that divisible by 4? And you can do
that in your head. That's 4 times 22. So this is divisible by 4. Now, let's see. 4 goes into 60 15 times. And then to go from 60 to 70,
you have to get another 10, which is not divisible by 4. So that's not divisible by 4. And you can even try to
divide it out yourself. 4 goes into 70,
let's see, one time. You subtract, you get a 30. 4 goes into 30 seven times. You multiply, then you subtract. You get a 2 right over
here as your remainder, so it is not divisible by 4. Now, let's move on to 5. Now, you're probably already
very familiar with this. If your final digit is a 5 or
a 0, you are divisible by 5. So this one is not
divisible by 5. This one is divisible by 5. You have a 0 there, so
this is divisible by 5. And this, you have a
5 as your ones digit. So once again-- finally-- this
is divisible by something. It's divisible by 5. Now, the number 6. The simple way to think
about divisibility by 6 is that you have to be
divisible by both 2 and 3 in order to be divisible by 6,
because the prime factorization of 6 is 2 times 3. So here, we're
divisible by 2 and 3, so we're going to be
divisible by-- let me do that in a new
color-- so we're going to be divisible by 6. Here, we're
divisible by 2 and 3, so we're going to
be divisible by 6. And if you were just divisible
by 2 or 3, just one of them, then you wouldn't
be able to do this. You have to have
both a 2 and a 3, divisibility by both of them. And here, you're divisible
by neither 2 nor 3, so you're not going
to be divisible by 6. Now, let's do the test for 9. The test for 9 is very
similar to the test for 3. Sum up all the digits. If that sum is divisible
by 9, then you're there. Well, we already summed
up the digits here, 48. 48 actually is not
divisible by 9. If you're not sure, you can
add up the digits there. You get 12. 12 is definitely
not divisible by 9. So this thing right over
here is not divisible by 9. And this one over here, if
you added up all the digits, we got 18, which is divisible. It is divisible by 9. And I'm running out of colors. So this one is divisible by 9. All the digits added up to 18. And this one over here, you
don't even have to add them up, because we already know
it's not divisible by 3. If it's not divisible by 3,
it can't be divisible by 9. But if you did
add up the digits, you get 19, which is
not divisible by 9. So this also is
not divisible by 9. And then finally,
divisibility by 10. And this is the easiest one
of all, because you just have to see if you have
a 0 in the ones place. You clearly do not have a
0 in the ones place here. You do have a 0 in
the ones place there, so you are divisible by 10 here. And then finally, you don't
have a 0 in the ones place here, so you're not going
to be divisible by 10. Another way you
could think about it, you have to be divisible by both
2 and 5 to be divisible by 10. Here, you are divisible
by 5 but not by 2. But obviously,
the easiest one is to just see if you have
a 0 in the ones place.