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# e-book Prime Factorization for the First 10,000 Numbers

Advanced search. Author Topic: Variables related question. Read times. RubenMG New member Posts: 5. Hi guys! Hope you're all doing well I have a question, i was trying to do the 3rd Euler project problem. Here i put you the code. What is the largest prime factor of the number ?

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Lazarus 2. It should be: 5, 7, 13 and Bart Hero Member Posts: First try to get the algorithm right. You only want to divide by prime numbers.

How to Find the Least Common Multiple Using Prime Factorization : Multiplication Tips

The array representing the even number 10 has the dots divided evenly into two equal rows of 5, but the array representing the odd number 11 has an extra odd dot left over. Indeed, our concept of the number 2 is so different from our conceptions of all other numbers that we even use different language. We divide a pie between two people, but among three people. We identify two alternatives , but three options. There are several obvious facts about calculations with odd and even numbers that are very useful as an automatic check of calculations.

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First, addition and subtraction:. Proofs by arrays usually convince students more than algebraic proofs. The other cases are very similar. Draw four diagrams to illustrate the four cases of subtraction of odd and even numbers. Proofs by arrays can be used here, but they are unwieldy. Instead, we will use the previous results for adding odd and even numbers.

Here are examples of the three cases:. The first and second products are even because each can be written as the sum of even numbers:. The third product can be written as the sum of pairs of odd numbers, plus an extra odd number:.

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• Each bracket is even, because it is the sum of two odd numbers, so the whole sum is odd. What can we say about the quotients of odd and even numbers? Assuming in each case that the division has no remainder, complete each sentence below, if possible. Justify your answers by examples. The previous results on the arithmetic of odd and even numbers can be obtained later after pronumerals have been introduced, and expansions of brackets and taking out a common factor dealt with. The important first step is:.

### Wrapping up

Obtain the previous results on the addition and multiplication of odd and even numbers by algebra. In the previous section, we represented even numbers by arrays with two equal rows, and odd numbers by arrays with two rows in which one row has one more dot than the other. Representing numbers by arrays is an excellent way to illustrate some of their properties.

For example, the arrays below illustrate significant properties of the numbers 10, 9, 8 and 7.

## Prime factors in java

The convention used in these modules is that the first factor represents the number of rows, and the second factor represents the number of columns. The opposite convention, however, would be equally acceptable. The first array shows that 9 is a square because it can be represented as a square array. Because there is no 2-row array, the number 9 is odd. The second array is the trivial array. The third array shows that 8 is a cube because it can be represented as a cubic array. The first array shows that 8 is even, and the second array is the trivial array.

The interesting thing about the number 7 is that it only has the trivial array, because 7 dots cannot be arranged in any rectangular array apart from a trivial array. Numbers greater than 1 with only a trivial rectangular array are called prime numbers. All other whole numbers greater than 1 are called composite numbers. We shall discuss prime numbers in a great deal more detail in the later module, Primes and Prime Factorisation.

The usual definition of a prime number expresses exactly the same thing in terms of factors:. Here are the only possible rectangular arrays for the first four prime numbers:. Rectangular arrays are not the only way that numbers can usefully be represented by patterns of dots. In the diagram to the right, two copies of the fourth triangular number have been fitted together to make a rectangle.

## Factors of an integer - Rosetta Code

Explain how to calculate from this diagram that the 4th triangular number is Hence calculate the th triangular number. Multiples, common multiples and the LCM. Because 6 is an even number, all its multiples are even. The multiples of an odd number such as 7, however, alternate even, odd, even, odd, even,… because we are adding the odd number 7 at each step.

The number zero is a multiple of every number. The multiples of zero are all zero. Every other whole number has infinitely many multiples.

We can illustrate the multiples of a number using arrays with three columns and an increasing numbers of rows. Here are the first few multiples of Rows and columns can be exchanged. Thus the multiples of 3 could also be illustrated using arrays with three rows and an increasing numbers of columns. The repeating pattern of common multiples is a great help in understanding division. Here again are the multiples of 6,. If we divide any of these multiples by 6, we get a quotient with remainder zero. For example,. To divide any other number such as 29 by 6, we first locate 29 between two multiples of 6.

Thus we locate 29 between 24 and Notice that the remainder is always a whole number less than 6, because the multiples of 6 step up by 6 each time. Hence with division by 6 there are only 6 possible remainders,. This result takes a very simple form when we divide by 2, because the only possible remainders are 0 and 1.

In later years, when students have become far more confident with algebra, these remarks about division can be written down very precisely in what is called the division algorithm. The table above shows all the whole numbers written out systematically in 7 columns. Suppose that each number in the table is divided by 7 to produced a quotient and a remainder.

When 22 and 41 are divided by 6, their remainders are 4 and 5 respectively. 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.