Given a range [n,m], find the number of elements that have odd number of factors in the given range (n and m inclusive).
Input : n = 5, m = 100 Output : 8 The numbers with odd factors are 9, 16, 25, 36, 49, 64, 81 and 100 Input : n = 8, m = 65 Output : 6 Input : n = 10, m = 23500 Output : 150
A Simple Solution is to loop through all numbers starting from n. For every number, check if it has an even number of factors. If it has an even number of factors then increment count of such numbers and finally print the number of such elements. To find all divisors of a natural number efficiently, refer All divisors of a natural number
An Efficient Solution is to observe the pattern. Only those numbers, which are perfect Squares have an odd number of factors. Let us analyze this pattern through an example.
For example, 9 has odd number of factors, 1, 3 and 9. 16 also has odd number of factors, 1, 2, 4, 8, 16. The reason for this is, for numbers other than perfect squares, all factors are in the form of pairs, but for perfect squares, one factor is single and makes the total as odd.
How to find number of perfect squares in a range?
The answer is difference between square root of m and n-1 (not n)
There is a little caveat. As both n and m are inclusive, if n is a perfect square, we will get an answer which is less than one the actual answer. To understand this, consider range [4, 36]. Answer is 5 i.e., numbers 4, 9, 16, 25 and 36.
But if we do (36**0.5) – (4**0.5) we get 4. So to avoid this semantic error, we take n-1.
Count is 8
Time Complexity : O(1)
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- Number of elements with even factors in the given range
- Find number of factors of N when location of its two factors whose product is N is given
- Sum of all even factors of numbers in the range [l, r]
- Sum of all odd factors of numbers in the range [l, r]
- K-Primes (Numbers with k prime factors) in a range
- Count numbers from range whose prime factors are only 2 and 3
- Count elements in the given range which have maximum number of divisors
- Count numbers in a range having GCD of powers of prime factors equal to 1
- Prime factors of LCM of array elements
- Minimum elements to be added in a range so that count of elements is divisible by K
- Queries on sum of odd number digit sums of all the factors of a number
- Number with maximum number of prime factors
- Sum of all the factors of a number
- Prime factors of a big number
- Product of factors of number
Improved By : nitin mittal