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Suppose that each number in the table is divided by 7 to produced a quotient and a remainder.
What is the same about the results of the division in each row? Common multiples and the LCM An important way to compare two numbers is to compare their lists of multiples. Let us write out the first few multiples of 4, and the first few multiples of 6, and compare the two lists.
The numbers that occur on both lists have been circled, and are called common multiples. The common multiples of 6 and 8 are 0, 12, 24, 36, 48,… Apart from zero, which is a common multiple of any two numbers, the lowest common multiple of 4 and 6 is These same procedures can be done with any set of two or more non-zero whole numbers.
A common multiple of two or more nonzero whole numbers is a whole number that a multiple of all of them. The lowest common multiple or LCM of two or more whole numbers is the smallest of their common multiples, apart from zero.
Hence write out the first few common multiples of 12 and 16, and state their lowest common multiple. Hence write down the LCM of 12, 16 and 24? Solution a The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96,,… The multiples of 16 are 16, 32, 48, 64, 80, 96,,… Hence the common multiples of 12 and 16 are 48, 96, ,… and their LCM is Two or more nonzero numbers always have a common multiple — just multiply the numbers together.
But the product of the numbers is not necessarily their lowest common multiple. What is the general situation illustrated here? Solution The LCM of 9 and 10 is their product The common multiples are the multiples of the LCM You will have noticed that the list of common multiples of 4 and 6 is actually a list of multiples of their LCM Similarly, the list of common multiples of 12 and 16 is a list of the multiples of their LCM This is a general result, which in Year 7 is best demonstrated by examples.
In an exercise at the end of the module, Primes and Prime Factorisationhowever, we have indicated how to prove the result using prime factorisation. This can be restated in terms of the multiples of the previous section: On the other hand, zero is the only multiple of zero, so zero is a factor of no numbers except zero.
These rather odd remarks are better left unsaid, unless students insist.
They should certainly not become a distraction from the nonzero whole numbers that we want to discuss. The product of two nonzero whole numbers is always greater than or equal to each factor in the product.
Hence the factors of a nonzero number like 12 are all less than or equal to Thus whereas a positive whole number has infinitely many multiples, it has only finitely many factors.
The long way to find all the factors of 12 is to test systematically all the whole numbers less than 12 to see whether or not they go into 12 without remainder.Edit Article How to Find the Least Common Multiple of Two Numbers.
In this Article: Article Summary Listing all Multiples Using Prime Factorization Using the Grid or Ladder Method Using Euclid’s Algorithm Community Q&A A multiple is the result of multiplying a number by an integer.
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You have probablry used computers to do all sorts of useful and interesting things. In each application, the computer responds in different ways to your input, from the keyboard, mouse or a file. Find the least quotient. 7, Find the greatest quotient. 2 4, Find the quotient closest to, Find a 3-digit quotient with a 4 in the ones place.
1, Find a quotient of 1, 2, 8, Find the least quotient that ends with a Find the quotient closest to 1, 4, E13 Grade 5. Sources. For most Unix systems, you must download and compile the source code. The same source code archive can also be used to build the Windows and Mac versions, and is the starting point for ports to all other platforms.
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