Factors and Multiples

Factors

       When we express a number as a product of two or more natural numbers, then each of these natural numbers is a factor of the given number.

       For example,

24=1 x 24
=2 x 12
=3 x 8
=4 x 6

       Each of the natural numbers 1, 2, 3, 4, 6, 8, 12, and 24 is a factor of 24.

       We can see that 24 is a divisible (can be divided exactly) by each of its factors. In general, the method of division can be used to test if a number is a factor of another number.

24 / 1=24      24 / 6=4

24 / 2=12      24 / 8=3

24 / 3=8      24 / 12=2

24 / 4=6      24 / 24=1
...

MATH FOCUS

  • All numbers are divisible by 1; therefore, 1 is a factor of any number.

  • All even numbers are divisible by 2; therefore, 2 is a factor of any even number.

  • Any number (except 0) can be divided exactly by itself; therefore, any number is a factor of itself.

Prime and Composite Numbers

       Every counting number, except 1, has atleast two factors, as the following list shows.

Counting Numbers Factors
18 1, 2, 3, 6, 9, 18
14 1, 2, 7, 14
23 1, 23
17 1, 17
21 1, 3, 7, 21
36 1, 2, 3, 4, 6, 9, 12, 18, 36
3 1, 3

       We are particularly interested in those numbers that have exactly two different factors. In the list 23, 17, and 3 fall into this category. They are called prime numbers.

MATH FOCUS

  • A prime number is a countinf number with exactly two different factors (or divisors).

  • A composite number is a counting number with more than two different factors (or divisors).

Prime Factorization

       Factorization, where all the factors are prime numbers, is called prime factorization.

       To find prime factors, we divide the number by prime number like 2, 3, 5, 7, ... until the quotient becomes 1.

Example:       Express 120 as a product of prime factors.

Solution

120=2 x 60 -- 120/2

=2 x 2 x 30 -- 60/2

=2 x 2 x 2 x 15 -- 30/2

=2 x 2 x 2 x 3 x 5 -- 15/3

=23 x 3 x 5 -- 5/5

       Therefore, 120=2 x 2 x 2 x 3 x 5=23 x 3 x 5.

Divisibility Rules

Test of Divisibility


If the number has this feature Then it is divisible by
The number ends with 0, 2, 4, 6, or 8. 2
The sum of its digits is divisible by 3. 3
The number represented by its last two digits is divisible by 4 (00 is considered to be divisible by 4). 4
The number ends with 0 or 5. 5
The number meets the tests for divisibility by 2 and 3 6
The difference when twice the last digit is subtracted from the remaining digits (without the last digit) is 0 or divisible by 7. The process may be repeated. 7
The number represented by its last three digits is divisible by 8. 8
The sum of its digits are divisible by 9. 9
The number ends in 0. 10
The difference of the sum of its digits in odd positions and the sum of its digits in even positions (starting from the unit's place) is either 0 or divisible by 11. 11
The number meets the test for divisibility by 3 and 4. 12
The number meets the divisibility by 3 and 5. 15
The last two digits from a number which is divisible by 25. 25

Combined Test for Divisibility by 7, 11, and 13

       We now describe a test for divisibility by 7, 11, and 13 all at the same time. Consider a number like

N=35,253,663,472,614,159,218,828.

       Break N up into the series of 3-digit numbers determined by the 3-digit groups startinf drom the right in N.

035       253       663       472       614       159       128       828

       Now, as in the test for divisibility by 11, compute the sum of the numbers in the odd-numbered positions in the above list and the sum of the numbers in the even-numbered positions. This gives

035

663

614

+218
1,530		 and 253

472

159

+828
1,712		   

as shown. The difference in these sums is 1,712 - 1,530=182 and N is divisible

by 7 if 182 is divisible by 7,

by 11 f 182 is divisible by 11, and

by 13 if 182 is divisible by 13.