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Prime Factors, HCF and LCM

Factors and Multiples

In this discussion we will only deal with the natural numbers (that is, positive integers): 1, 2, 3, 4, 5, …

If m = d×q we say d is a factor of m, and m is a multiple of d.

Note: d divides m so d is also called a divisor of m.

Since m = 1×m, 1 and m are both factors of m. The number 1 is called the trivial factor as every number is divisible by 1. m is called the improper factor as every number is also divisible by itself.

A number that has a nontrivial proper factor is said to be composite.

Remark: Suppose m=d*q and d is greater than or equal to sqrt(m)then q is less than or equal to sqrt(m) . Thus to discover the factors of a number m we need only test with numbers less than or equal to square root m.

Example

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

1, 2, 3, 4, 6 and 12 are factors of 12. 12 is a multiple of 1, 2, 3, 4, 6 and 12. 1 is a trivial factor. 12 is an improper factor. 2, 3, 4 and 6 are nontrivial proper factors so 12 is composite.

Exercise

Find a non-trivial proper factor of the given number. If none exists, enter none.

Number
Factor

 

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