Factors and divisors

The terms factor and divisor are used interchangeably. They are different ways of expressing the same mathematical truth and this is because of the inverse relationship between division and multiplication.

Divisors

For a number \(n\), its divisor is any number that divides \(n\) evenly without remainder: $$ \frac{a}{b} = 0 $$

In this operation, \(a\) is the divisor, \(b\) is the dividend and \(0\) is the quotient.

Factors

For a given number \(n\), its factors are any pair of numbers that when multiplied together return \(n\) as the product: $$ a \cdot b = n $$

We can see the relationship consists in the fact that factors are associated with multiplication and divisors are associated with division: two different perspectives on the same number relationships.

For example, 6 is both a factor and divisor of 18 and 24. To be precise, it is the greatest common divisor of these two numbers.

As a divisor:

$$ \\frac{18/6}{24/6} = \frac{3}{4} $$

As a factor:

$$ \\frac{3 \cdot 6}{4 \cdot 6} = \frac{18}{24} $$

When we divide by the common divisor is acts as a divisor. When we multiply by the common divisor it acts as a factor. The fact that the fractions are equivalent in both cases indicates that the properties are equivalent.

Greatest common divisor

For two two integers \(a, b\), \(D\) is a common divisor of \(a\) and \(b\) if it is a divisor of both. The greatest common divisor is the largest value that \(D\) can be whilst remaining a divisor to both \(a\) and \(b\).

Demonstration

Find the greatest common divisor of \(18\) and \(24\)

The divisors of 18: $$1, 2, 3, 6, 9, 18$$

The divisors of 24: $$ 1, 2, 3, 4, 6, 8, 12, 24$$

Thus the common divisors are: $$ 1, 2, 3, 6 $$

The largest value in the above set is 6, thus 6 is the greatest common divisor.

Heuristics for finding divisors

  1. For dividend \(n\) , if \(n\) ends in an even number or zero, \(n\) is divisible by 2.
    1. \(\frac{12}{2} = 6\)
    2. \(\frac{84}{2} = 42\)
  2. For dividend \(n\) if the sum of the digits is divisible by 3 then \(n\) is divisible by 3.
    1. \(\frac{72}{3} = 24\)
    2. \(\frac{21}{3} = 7\)
  3. For a dividend \(n\) if the number represented of the last two digits of \(n\) divides by 4 then \(n\) is divisible by 4
    1. \(\frac{324}{4} = 81\)
    2. \(\frac{532}{4} = 133\)
  4. For a dividend \(n\), if the last digit of \(n\) is divisible by 0 or 5, then \(n\) is divisible by 5.
    1. \(\frac{25}{5} = 5\)
  5. For a dividend \(n\), if \(n\) is divisible by 2 and 3, then \(n\) is divisible by 6.
    1. \(\frac{12}{6} = 2\)
    2. \(\frac{18}{6} = 3\)
  6. For a dividend \(n\), if the last three digits of \(n\) are divisible by 8, then \(n\) is divisible by 8.
    1. \(\frac{73024}{8} = 9128\)
  7. For a dividend \(n\), if the sum of the digits of \(n\) is divisible by 9 then \(n\) is divisible by 9.
    1. \(\frac{117}{9} = 13\)
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