What does prime and proper mean?

Answer 1

The phrase prime and proper is generally used in connection with factoring of natural numbers. To explain its meaning we may need to sneak up on it.

• a natural number is a whole number greater than zero : 1,2,3,4,5, etc
• a factor of a natural number is another natural number which divides the first one evenly (with no remainder); e.g. 3 and 5 are factors of 15.
• every natural number can be divided by 1, and also by itself. Since this is always true, it is trivial to mention it; these factors are called improper.
• A proper factor is a natural number that divides the number being factored, but is not equal to the number being factored nor to unity.
• a Prime number is one with no proper factors.
• "prime and proper" factors are those proper factors which are also prime.
• When mathematicians want to factorise a number they are mostly interested in prime and proper factors. Improper factors are too obvious to mention. As to prime factors, they are what are needed, and may be hard to find. Once you know that the prime and proper factors of 12 are 2,2 and 3 that is all that is needed. To list non-prime factors like 4 and 6 is fairly trivial; anyone can see that just by glancing at the prime and proper factors.
• There is one good reason for listing the non-prime factors; when children are being taught to handle numbers, especially multiplication and division, asking for all the proper factors is good practice and helps the teacher to spot those who need extra help. Thus, the factors of 12 are noted as 2,3,4,6, even though 4 and 6 are not prime. Whether to include 1 and 12 as factors may be controversial amongst teachers of very young students.
• Older students should adopt the habit of giving only prime and proper factors as soon as they can.

Answer 2

If you're talking about prime and proper numbers, here we go.

Prime number: A number only divisible by itself and one (1)

Proper fraction: A fraction that has a bigger numerator than the denominator i.e 1/2