All Factors in Sorted Order

Given a natural number n, print all distinct divisors in sorted order.

Examples:

Input : n = 10
Output: 1 2 5 10

Input: n = 100
Output: 1 2 4 5 10 20 25 50 100

Input: n = 125
Output: 1 5 25 125

We have discussed how to find all the divisors of a number here:All Factors or Divisors

Using Two Arrays - O(√n) Time and O(√n) Space

• Check factors up to √n because factors come in pairs: n = a × b. When a is small (≤ √n), its pair b = n / a is large (≥ √n). So by iterating i from 1 to √n, we find one of the two factors as i and n/i.
• When we iterate through factors, the smallest factor i gives the largest pair (n/i) as smaller value of i means larger value (n/i).
• Storing smaller factors first and then adding larger factors in reverse gives all divisors in sorted order without sorting.
• We need to handle perfect squares explicitly as we need to print all divisors only once.

1 2 4 5 10 20 25 50 100 

Constant Extra Space - O(√n) Time and O(1) Space

This is mainly a space optimization over the above approach. We first get the smaller factor of pairs in sorted order traversing i to √n. To get the larger factors in increasing order, we traverse back from i - 1 (which is around √n - 1) to 1 and instead of storing i, we store n/i so that we larger elements of pairs in increasing order.

For example, let n = 10.

Step 1: Loop from 1 → √n (√10 ≈ 3, so check only 1, 2, 3)

• i = 1 ->10 % 1 == 0 -> print 1
• i = 2 -> 10 % 2 == 0 -> print 2
• i = 3 -> not divisible -> ignore

Step 2: i = i - 1, i = 2

Step 2: Traverse from 2 to 1

• i = 2 -> 10 / 2 = 5 -> print 5
• i = 1 -> 10 / 1 = 10 -> print 10

Final Output : 1, 2, 5, 10

1 2 4 5 10 20 25 50 100