Canonical Representation of a Number

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 = p₁^k1xp₂^k2x …x p_m^km, where p₁,p₂,…,p_m are primes where 1<p₁<p₂<…<p_k and k₁,k₂,…,k_m 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 = p₁^k1xp₂^k2x …x p_m^km is the canonical representation of a number n then the number of positive divisors of n is given by (1+k₁)(1+k₂)…(1+k_m).

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 = p₁^k1xp₂^k2x …x p_m^km 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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