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How do you find the greatest odd divisors of a number?
5 Answers
- f(2k+1) = 2k+1, i.e. the largest odd divisor of an odd number is the number itself.
- f(2k) = f(k). i.e the largest odd divisor of an even number 2m is same as the largest odd divisor of the number m.
- Sum of first k odd numbers is equal to k^2.
How do you find the number of odd divisors?
Now we are going to find the number of odd divisors. For this, we will multiply the power of only odd prime factors by adding $ 1 $ . Therefore, number of odd divisors $ = \left( {b + 1} \right)\left( {c + 1} \right) = \left( {1 + 1} \right)\left( {2 + 1} \right) = 2 \times 3 = 6 $ .
What types of numbers will have an odd number of divisors?
We can observe that the number of divisors is odd only in case of perfect squares. Hence the best solution would be to check if the given number is perfect square or not. If it’s a perfect square, then the number of divisors would be odd, else it’d be even.
How many odd divisors does the number 1000000 have?
It have 7 odd divisors in it.
Can you find all numbers between 1 and 100 which have an odd number of factors?
The composite odd numbers from 1 to 100 are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, and 99.
Do all odd numbers have odd factors?
Only those numbers, which are perfect Squares have an odd number of factors. For example, 9 has odd number of factors, 1, 3 and 9. 16 also has odd number of factors, 1, 2, 4, 8, 16.
How many positive divisors does 9000 have?
The number 9,000 can be divided by 48 positive divisors (out of which 36 are even, and 12 are odd).
How to find the greatest odd divisor of a number?
Given an interval of integers [A, B]. For each number in this interval compute its greatest odd divisor. Output the sum of these divisors. Recommended: Please try your approach on {IDE} first, before moving on to the solution.
How to find the answer for an odd number?
We need to find the answer in range [ A, B ] if we can find answer in range [ 1, B ] and subtract it from [ 1, A -1 ] then we will get our required answer. The answer for an odd number X is X itself. The answer for an even number X is equal to the answer for X/2.
Can a number with odd numbers of divisors be a perfect square?
Integers with odd numbers of divisors are always perfect squares. Given some number N, if X is a divisor there must be some corresponding Y such that XY=N holds true. They always come in pairs. The exception is a perfect square.
How to find the number of factors an integer has?
Those powers can be used to get the number of factors an integer has. If the number is num and it can be represented as (a p1) * (b p2) * (c p3 ) Below is the implementation of the above approach: