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A prime number is a number greater than 1 that can only be divided exactly by itself and by 1. If a number can be divided exactly by any other number that is not itself or 1, it is not prime and is called compound number.
Divisors and multiples
Students need to know what is a divisor and what is a multiple to study prime numbers. These two types of numbers are often confused. A divisor is a number that exactly divides a certain number. A multiple is a number that results from multiplying a certain number by another integer.
Prime numbers are integers that must be greater than one; therefore 0 and 1 are not considered prime numbers, nor are any numbers less than zero. The number 2 is the smallest prime number, since it meets its definition: it can only be divided by itself and by 1.
The factorization method to identify a prime number
You can quickly determine if a number is prime by factoring it or breaking it down into its prime factors. Factoring a number consists of identifying its prime divisors, a divisor being an integer number that can be multiplied by another to obtain the original number.
For example, if we consider the number 10, the numbers 2 and 5 are divisors of 10 since each of them is an integer that can be multiplied by another to obtain the result 10. At the same time, 1 and 10 are also divisors of 10. Furthermore, 2 and 5 are prime numbers, and are then the prime factors of the number of the number 10, since both 1 and 10 are not prime numbers, and 2 and 5 then constitute the factorization or decomposition into prime factors of the number 10 Thus we see that the number 10 has factors other than itself and the number 1, so 10 is not a prime number.
An easy way for students to use factoring to determine if a number is prime is to give them concrete items to count, such as buttons or coins, that represent a certain whole number. They can then divide them into smaller groups and identify if those smaller groups that compose it are repeated and thus constitute a divider. For example, they might divide 10 buttons into two groups of five or five groups of two.
Factoring or prime factorization of a number can be done by determining the factors sequentially. For example, if you want to divide the number 30 into prime factors, you could start with 10 x 3 or 15 x 2. In each case, continue factoring each of the components until you get only prime factors; in this case 10 (2 x 5) and 15 (3 x 5). The final result will produce the same prime factors since the prime factorization of a number is unique. In this example it is 2, 3 and 5, because 5 x 3 x 2 = 30, as is 2 x 3 x 5.
using a calculator
After using the method described in the previous section, students can use a calculator and apply the concept of divisibility to determine if a number is prime.
To determine if a number is prime, the student can enter the number into the calculator and see if it can be divided evenly by some whole number less than the original number. If we consider, for example, the number 57, we can try to divide it by 2 and we will see that the quotient is 28.5, which is not an integer. But dividing it by 3 will get the number 19; therefore 19 and 3 are divisors of 57 different from 1 and 57, thus showing that 57 is not a prime number.
Simple pencil and paper division can also be a good way to teach youngsters how to determine prime numbers. The number in question is divided first by 2, then by 3, then by 5 and so on with the following prime numbers until we reach the number we are studying. If the result of dividing by the smallest prime numbers does not produce an integer in any case, then the number in question is prime. This simple method is useful in helping the student understand what makes a number prime.
