Prime and composite numbers:
An integer is called prime if it is divisible only by 1 and itself. i.e. a prime number cannot be divisible by any other number than 1 and itself. A prime number has exactly 2 positive divisors. A number which has more than 2 positive divisors is said to be a composite number. i.e. a composite number contains at least one divisor other than 1 and itself. Here we need to make a note that 1 is neither prime nor composite.
Canonical representation of a number:
Every integer can be expressed as the product of primes and this factoring of the integer into primes is unique except the order of primes.
So, any integer n can be expressed as, n = p1k1xp2k2x …x pmkm, where p1,p2,…,pm are primes where 1<p1<p2<…<pk and k1,k2,…,km are positive integers.
Number of all positive divisors of a number:
If we take a number 6, we can say that there are only 4 divisors for 6 which are 1,2,3,6. Similarly if we take a number 24 we say that there are 8 divisors namely 1,2,3,4,6,8,12,24. But what if we have to find the number of divisors of a big number such as 2000. There is a formula for finding the number of divisors of a given number.
If n = p1k1xp2k2x …x pmkm is the canonical representation of a number n then the number of positive divisors of n is given by (1+k1)(1+k2)…(1+km).
Example:
2000 = 24 x 53
Number of positive divisors of 2000 = (1+4)(1+3) = 5×4 = 20.
Sum of all positive divisors of a number:
As we discussed earlier, when it is difficult to find the number of divisors, it is obviously more difficult to find the sum of divisors. There is a formula for finding the sum of divisors of a given number.
If n = p1k1xp2k2x …x pmkm is the canonical representation of a number n then the sum of positive divisors of n is given by
Example:
2000 = 24 x 53
Sum of positive divisors of 2000 =[25-1/2-1][54-1/5-1] = [31/1] x [624/4] = 31 x 156 = 4836.
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