What is the smallest 4-digit number that is divisible by 2,3,4, 5, 6, 8,9 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.

What is the smallest 4

Which is the smallest 4

What is the smallest 4

What is the smallest 4