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Divisibility Rules

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Divisibility Rules. So first of all, let's start up with a couple easy questions. Is 56 divisible by 7? Is 50, divisible by 13? Theoretically, those are questions you should be able to answer with little problem.

The answer to the first of course, is yes, 56 equals 7 times 8. The answer to the second of course is no, 13 does go into 52 but it does not go into 50. So first of all I'll just mention, if these are not questions that you can answer just by looking at them, probably you need to practice your times tables a little bit more.

You really need to have your basic times tables down so that questions like this are in fact very easy. Now, a slightly harder question is that long number divisible by 3. Well, no one expects you to do this division in your head. We can't do the full division of find the quotient, but we can use it the divisibility rule to answer the question, and the test loves divisibility roles.

So first of all, divisibility by 2. Of course, all even numbers that are divisible by 2. To tell whether a large numbers even, all we have to do is look at the last digit. If the one's digit is even, then the number is even. So we have these large numbers, we can ignore the rest of the number and just look at the one's place.

And in the one's place we see 5, 1, 7, 3, those are all odd, but the 6 is even. That means that the middle one was like one digit is 6, that's the only even number, the only number divisible by 2. So that's divisibility by 2. Divisibility by 5. This is another divisibility rule that involves looking at the last digit.

If the last digit is a 5 or 0, then the number is divisible by 5, otherwise, it isn't. So again we have big numbers like these, we can ignore the rest of the number and just look at that last digit. In deed we have a last digit of 5, so that number is divisible by 5, but the others do not end in 5 or 0, so they're not divisible by 5.

So both of those rules, divisibility by 5 and divisibility by 2, those just involve looking at the one's place and nothing else. Now, the Rule for 4. This rule is similar, here, we look at the last two digits, the ten's place and the one's place, so we have to look at two digits. If the last two digits for me to digit number divisible by 4, then the entire number is divisible by 4.

So again, our same list of long numbers, look at those last two digits and think of them as two digit numbers, 55, 41, 96, 37, 33. The only one amongst those that is divisible by 4 is 96, 96 is a number divisible by 4. So that means that hole, middle number is divisible by four. Now, divisibility for 3, the test loves this rule.

This rule is a little different. Here, we add up all the digits of the number. If the sum of the digits is divisible by 3, then the number is divisible by 3 and if the sum of the digits is not divisible by 3, the number is not divisible by 3. So for example with 135 we add 1 plus 3 plus 5, that's 9, so 9 is divisible by 3, 135 must be divisible by 3.

With 734, we add 7 plus 3 plus 4 that equals 14. Since 14 is not divisible by 3, we know that 734 is not divisible by 3. 1296, we add 1 plus 2 plus 9 plus 6, that equals 18. And since 18 is divisible by 3, then 1296 must be divisible by 3. So it works the same way for large numbers.

Here's the question we had at the very beginning of the module. So is that large number divisible by 3? Well, here's what I noticed, I noticed that middle 3 plus 3 plus 4 that equals 10. I can take the 5 plus 5 that equals 10. And that only leaves a 1, a 0, a 2 and another 1, and those add up to 4.

So that means that everything adds up to 24. The sum of the judges is 24, which is divisible by 3, so the original number must be divisible by 3. So all we have to do is add up of the digits, and then see if that's divisible by 3 and then that tells us whether the number overall is divisible by 3. The Divisibility Rule for 9.

This is exactly like the rule for 3. Add all the digits, if the sum of the digits is divisible by 9, then the number is divisible by 9. If the sum of the digits is not divisible by 9, the number is not divisible by 9. So for example 1296 we found in the last few slides that the sum of this was 18. The sum is divisible by 9, so 1296 must be divisible as well.

3072, we add these digits up we get a sum of 12. So the sum of the digits is divisible by 3 but not 9, so the number is divisible by 3 but not 9. Notice incidentally, 3072, the last two digits number, 72 is divisible by 4. So that's a number that would be divisible by 3 and by 4, which would mean it's divisible by 12.

We can also use the divisibility rule for 9, for larger numbers. So is this large number divisible by 9? Well, we already found out sum of the digits is 24. So 24 is divisible by 3 but not by 9. So the original number, that nine digit number is divisible by 3, but not by 9. The Divisibility Rule for 6.

Here, we're gonna have a combination. In order to be divisible by 6, a number must be, a, divisible by 2, and b, divisible by 3. We checked divisibility by 2 by looking at the last digit, making sure that it's even. And we checked the visibility by 3 by finding the sum of the digits.

So any even number divisible by 3 has to be divisible by 6. Is 1296 divisible by 6? Well, first of all, we know what's even, the sum of the digits we found already is 18, which is divisible by 3. Therefore 1296 is divisible by 6. Is this long number divisible by 6?

Well, clearly it's even, that's easy to determine. We add the digits at the first three that up to 15. The second three digits at up to 14. The last three digits at up to 17. 15 plus 17 plus 14 is 46. The sum of the digits is not divisible by 3, so the original number is not divisible by 3 or by 6.

In this video we discuss the most common divisibility rules, the rules for 2, 5, 4, 3, 9 and 6.

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