#MathsInMinutes Day 13:

Day 13: Divisors and remainders

A number is a divisor of another number if it divides into that number  exactly, with no remainders. So 4 is a divisor of 12, because it can be divided into 12 exactly three times. In this kind of operation, the number being divided, 12, is known as the dividend.

But what about 13 divided by 4? In this case, 4 is not a divisor of 13, since it  divides into 13 three times, but leaves 1 left over. One way of expressing the answer is as three, remainder one. This is another way of saying that 12, which is 3 ∙ 4, is the largest whole number less than the dividend (13) that is divisible by four, and that 13=12+1. 

When the remainder of one is now divided by four, the result is the fraction ¼,  so the answer to our original question is 3¼.

3 and 4 are both divisors of 12 (as are 1, 2, 6 and 12) If we divide one natural  number, p say, by another, q, that is not a divisor of p, then there is always a remainder, r, that is less than q. This means that in general p = kq + r, where k  is a natural number, and r is a natural number less than q.

For any two numbers p and q, the greatest common divisor, GCD, also known  as the highest common factor, is the largest number that is a divisor of both p and q. Since 1 is obviously a divisor of both numbers, the GCD is always  greater than or equal to 1. If the GCD is 1, then the number is said to be coprime – they share no common positive divisor except 1.

Divisors give rise to an interesting family of numbers called 'perfect numbers'.  These are numbers whose positive divisors, excluding themselves, sum to the value of the number itself. The first and simplest perfect number is 6,  which is equal to the sum of its divisors, 1, 2 and 3.

The second perfect number is 28, which is equal to 1+2+4+7+14.

You have to wait a lot longer to find the third: 496, which is equal to 1+2+4+8+16+31+62+124+248.

Perfect numbers are rare, and finding them is a challenge.

Mathematicians have yet to find conclusive answers to some important  questions, such as whether there are an infinite number of perfect numbers, or whether they are all even.

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