The dividend (10) is also marked and you see that it’s not one of those multiples. These are the products I was talking about. Notice that the first five positive multiples of 3 ( d) are marked. This all sounds very abstract, so let’s see how it works in the example above: And we call the k which satisfies this requirement the quotient ( q). The integers divisible by d are all in the form for an arbitrary integer k. The idea is to to find the greatest integer less than or equal to D that is divisible by d without remainder (that is, with remainder equal to 0). Let’s consider any division, where D is the dividend and d is the divisor. Now let’s formalize this procedure a little bit so we can generalize it to the whole set of integers. Then, our quotient became 3 (because ) and our remainder became 1 (because ). How exactly did we find these values? Well, out of the 10 apples, we took 9 and split them into groups of 3, right? We took 9 because this is the largest number divisible by 3 that is still less than the dividend (10). You can divide 9 of them into groups of 3 (the quotient) and the last apple will be the remainder. For example, say you and two of your friends want to divide 10 apples equally between the three of you. I already gave you the basics of this method in my post on natural numbers. ![]() This is an early method used by the great Ancient Greek mathematician Euclid of Alexandria that we still use today (after some evolution). Alternative conventions for calculating the quotientĮuclidean division (also known as division with remainder) is a special type of division that returns two numbers. ![]()
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