
Divisibility theorems and fraction flippingSeptember 21, 2019 Contents
Divisibility theoremsWhen it comes to divisibility, there exist some neat theorems to test a certain number’s different divisibilities, or factors. The following are those theorems and their proofs. Some of the proofs are more trivial than others, such as the proof for divisibility by 1, 2, 5, and 10. From here on out, it’s assumed that every number, when working with divisibility, is an integer. What is the smallest number that is divisible by 1 through 10? The answer is 2520. You can try the different divisibility theorems below to prove it. Divisibility by 1This theorem is quite easy to remember. Every integer is divisible by 1. Divisibility by 2It’s common knowledge that every even number (numbers ending with an even number) is divisible by 2. This is because even numbers are multiples of 2. In short, if a number ends with either 0, 2, 4, 6, or 8, it is divisible by 2. Say we have a fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. If d is divisible by 2, we can represent it as an even number 2n, like so: We can now see that the number is divisible by 2 if, and only if, the last digit is divisible by 2. And so the theorem is proven. Divisibility by 3The theorem goes that if the sum of all digits in a number is divisible by 3, the whole number is divisible by 3, i.e. Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s break out one a, b, and c from the first 3 terms, and factor out 3 like so: We can now see that the first term is divisible by 3, and the second term is divisible by 3 if, and only if, the sum of a + b + c + d is divisible by 3. And so the theorem is proven. We have shown that the procedure above will hold for all cases. The procedure is also recursive. If the sum is too hard to test for divisibility, the procedure can be repeated until a smaller sum is revealed. This is rarely necessary as the divisibility of the sum is often easy to determine. Divisibility by 4The theorem goes that if the last two digits of a number are divisible by 4, the whole number is divisible by 4, i.e. Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s factor out 4 from the first two terms, like so: We can now see that the first term is divisible by 4, so the whole number is divisible by 4 if, and only if, the second term is divisible by 4. And so the theorem is proven. Divisibility by 5The theorem goes that if the last digits of a number are divisible by 5, the whole number is divisible by 5, i.e. Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s factor out 5 from the first three terms, like so: We can now see that the first term is divisible by 5, so the whole number is divisible by 5 if, and only if, the second term is divisible by 5. And so the theorem is proven. As the only onedigit numbers that are divisible by 5 are 0 and 5, another way of putting it is that if the last digit is 0 or 5, the number is divisible by 5. Divisibility by 6This theorem is a combination of the theorem for divisibility by 2 and divisibility by 3. Note: Be careful when combining theorems like this. Don’t be fooled and try to combine the divisibility theorems for 2 and 4 to get the theorem for 8, as numbers divisible by 4 are always divisible by 2. The number 4 is, for instance, both divisible by 2 and by 4, but not by 8. Make sure the theorems you combine doesn’t share a factor, like 4 and 2 sharing factor 2. It’s, of course, possible to use theorems 4 and 2 together, but it’s not certain in all cases that the two theorems prove divisibility by 8. Divisibility by 7Probably one of the most useful theorems is the theorem of divisibility by 7, as it is recursive (just like the theorems of divisibility by 3 & 9). The theorem states that if the difference between the last digit multiplied by 2 and the remaining digits in a number is divisible by 7, the whole number is divisible by 7. I’ll show you the procedure with an example below: Neat! So how and why does it work? For simplicity’s sake, we use a twodigit number ab, represented as 10a + b. The theorem says that if (A) 7  a  2b then (B) 7  10a + b. Let’s prove it! In order to prove the theorem, we must prove both A and B. So let’s start with A. If we have a  2b, and it’s divisible by 7, we know that 7 must be a factor of the expression. We can now create an equation. Multiply the whole equation by 10, and add one extra b: Now add 20b to each side of the equation, and try to factor out 7: We can now see that the right side of the equation is divisible by 7, and our left side says 10a + b. Neat, we now know that the twodigit number ab is divisible by 7. Now we must show that B implies A. That is, if a  2b is divisible by 7, then 10a + b is divisible by 7. Let’s prove B. Just as for A, we know that 7 must be a factor of the expression. We can now create another equation: Subtract 21b from the whole equation, and factorize: We can now see that the right side of the equation is divisible by 7, and on our left hand, 10 is not divisible by 7, so the expression inside the parenthesis must be. But isn’t that expression a  2b? We now have the proof for the theorem and can conclude that, indeed, We have shown that the procedure above will hold for all cases. Divisibility by 8The theorem is quite similar to the theorem for divisibility by 4. The theorem goes that if the last three digits of a number are divisible by 8, then the whole number is divisible by 8, i.e. Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s factor out 8 from the first term, like so, We can now see that the first term is divisible by 8, so the whole number is divisible by 8 if, and only if, the second term is divisible by 8. And so the theorem is proven. Divisibility by 9Much like the theorem for divisibility by 3, the theorem goes that if the sum of all digits in a number is divisible by 9, the whole number is divisible by 9, i.e., Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s break out one a, b, and c from the first 3 terms, and factor out 9 like so: We can now see that the first term is divisible by 9, and the second term is divisible by 9 if, and only if, the sum of a + b + c + d is divisible by 9. And so the theorem is proven. We have shown that the procedure above will hold for all cases. The procedure is also recursive. If the sum is too hard to test for divisibility, the procedure can be repeated until a smaller sum is revealed. This is rarely necessary as the divisibility of the sum is often easy to determine. Divisibility by 10The theorem goes that if the last digits of a number are divisible by 10, the whole number is divisible by 10, i.e. Why is this proposition true? Let’s use the fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d. This number can be represented as 1000a + 100b + 10c + d. Now let’s factor out 10 from the first three terms, like so: We can now see that the first term is divisible by 10, so the whole number is divisible by 10 if, and only if, the second term is divisible by 10. And so the theorem is proven. As the only onedigit number that is divisible by 10 is 0, another way of putting it is that if the last digit is 0, the number is divisible by 10. Divisibility by 11The theorem goes that a number is divisible by 11 if, and only if, the alternate sum of its digits is divisible by 11. Like so, Neat! So how and why does it work? For simplicity’s sake, we use a fourdigit number abcd, where a represents the number of thousands, b the number of hundreds, c the number of tens, and the number d This number can be represented as 1000a + 100b + 10c + d. This expression can also be represented in another way by manipulating the terms. We give and take in an alternating fashion, like so: Now we factorize the expression, like so: We can see that the first term in the expression is divisible by 11. This means that if, and only if, the sum of the other terms is divisible by 11, then the whole expression is divisible by 11, and so the theorem is proven. We have shown that the procedure above will hold for all cases, as the number can be extended with infinite digits and still follow the same pattern. FractionsThe fraction flip when dividingMany of you have probably been taught the trick of flipping the right fraction in a division to instead use simpler multiplication. This is how it works: We start with the division: From there, we can reconstruct the two fractions as the dividend over the divisor with a horizontal line, for simplicity’s sake, like so: After that, the "trick" can begin. First, we multiply both the dividend and the divisor with the inverse of the divisor. As long as we treat the dividend and the divisor the same way, this is fine. However, keep PEMDAS in mind! If any of the dividend or the divisor would have been an addition or subtraction, you would have to multiply both terms by x, either by x(a + b) or xa + xb. As you can see, the right fraction has now "flipped", and not by magic, but with logic and reason. So, as division by 1 is equal to the dividend, we can then solve the expression, like so: As the final cherry on top, we can prove the procedure by dividing 1 by 2, as we know this should result in one half. 