[MUSIC] So now we are in a position to prove several divisibility tests. And let's start with the following problem. Consider the number 3,756 and let's try to divide them by 10. What is the remainder and what is the quotient of this number? And note that the decimal system is convenient here. Indeed we can consider our number and we can break into two parts, 3,750 and 6. And then the first number is divisible by 10. Now our number is equal to 375 x 10 + 6. And note that this is exactly the expression of division of our number by 10 with the remainder. And so the remainder is 6 and the quotient is 375. So, in general, we have the full length, so, note that this argument applies to any number and in general we have the full length. Suppose we have some number and we divided by 10 with a remainder, then the remainder will do just the last digit of our number and the quotient is the number formed by digit of our initial number, except the last digit. So we have the full simple observation and not that it gives us a very convenient, but also simple corollary. If you have some integer number e and it is divisible by 10, if it's last digit is 0, right. So, note that 10 divides e even only if a remainder is 0 and note that the remainder is exactly, as we observed, the remainder is exactly the last digit. So, 10 divides e, even if the last digit is 0. Okay, now let's consider divisibility by 5. Okay, let's again start to do the problem, suppose we have the number 7,347 and the question is, is it divisible by 5? And we can use the same trick here, let's break our number into two parts, 7,347 = 734 x 10 + 7. And now we can break it even more and note that, okay, it is equal to (734 x 2) x 5 + 5 + 2. And note that the first two summons are divisible by five and the last one is not. And note that again, it is the expression for division of our number by 5 the remainder and here the remainder is 2. And since the remainder is 2, then our number is not divisible by 5. Okay and again, we can generalize it. If you have any integer number it is divisible by 5, if and only if, it's last digit is 0 or 5. Okay, so let's give a proof, let's just denote the last digit of a by b. And let's subtract b from a. And the resulting number has the last digit 0 and so it is divisible by 10 and so it is divisible by 5. So, a minus b is divisible by 5. And so, as we already shown before, this means that a and b have the same reminder when we divided them by 5. So the reminder of a is 0, if and only if, the reminder of b is 0, so it is divisible by 5, if and only if, b is divisible by 5, but b is divisible by 5 if b is a digit, so it's from 0 to 9, it is divisible by 5, if and only if, it is 0 or 5. So, we have shown our lemma. Some integer is divisible by 5, if and only if, it's last digit is 0 or 5, so it's very easy to check whether a given integer is divisible by 5. Okay and let's also consider divisibility by 2, a very similar idea applies here. An integer is divisible by 2, if and only if, its last digit is 0, 2, 4, 6, or 8. In other words, it is divisible by 2, if and only if, its last digit is divisible by 2. And again the proof is completely the same, so let's just denote the last digit of a by b, let's subtract b from a. And now we have a number, we have a last digit 0, it is clearly divisible by 2. And so this means that a and b has the same remainder when we try to divide them by 2. And so is divisible by 2, if and only if, b is divisible by 2 and b is divisible by 2 if it is either 0 or 2 or 4 or 6 or 8. So let's review what we have learned in this lesson. We are studying number theory and number theory deals with integer numbers and operations on them. Number theory is important for fast numerical computations and also, it is vital for modern cryptography, so this is an important area. And we have discussed some basic notions divisibility and remainders and we will use them later on to build more advanced theory. [MUSIC]