7!

Number Theory Level 3

Find the number of odd (positive) divisors of $7!$ .

The answer is 12.

3 solutions

Paul Ryan Longhas
Feb 23, 2015

The prime factorization of $7!$ is $2^4* 3^2*5*7$ . Each odd factor of $7!$ is a product of odd prime factors, and there are $3*2*2 = 12$ ways to choose the three exponents.

How? 7!= 7 * 6 * 5 * 4 * 3 * 2 * 1 We have 4 prime odd divisos (incl 1) or 3 prime odd divisors (excl 1)

4+ (combination of 2 among 3) + (combination of 3 among 3) = 4+3+1 = 8

or 3+(combination of 2 among 3) + (combination of 3 among 3) = 3+3+1 = 7

But forgot the 3 in 6. :D :P

Ananya Aaniya - 5 years, 1 month ago

There are also 12 negative odd divisors.

Doug Gwyn - 4 years, 4 months ago

Yes, I also marked 24 as the answer and had it marked wrong. If it is limited to positive divisors, the problem should so state.

Thomas Raffill - 3 years, 10 months ago

Ken Ken
Apr 14, 2015

7! = 2^4 × 3^2 × 5 × 7. To find odd divisors of 7!, ignore the powers of 2 and then calculate the number of combinations of powers of 3, 5, and 7.

The number of combinations = 3 × 2 × 2 = 12.

Therefore, there are 12 odd divisors of 7!.

Can you please explain why number of combination are 3×2×2?

Yajur Phullera - 3 years, 9 months ago

Nvm, got it.

Yajur Phullera - 3 years, 9 months ago

The odd divisors of 7! are 3 5 7 9 15 21 35 45 63 105 315 and ? I don't think 1 is one of the twelve so what number am I missing?

Brenda Kock - 3 years, 7 months ago

I don't think you are missing any, There are only 11 odd divisors, The list of divisors is: 3, 3x3, 5, 5x3, 5x3x3, 7, 7x3, 7x3x3, 7x5, 7x5x3, 7x5x3x3.

Mark Taylor - 3 years, 4 months ago

1, Because 1 is also odd divisor

Ryan Muhammad Alex - 2 years, 11 months ago

1 is one of the twelve, because 1 is also odd number

Ryan Muhammad Alex - 2 years, 11 months ago

Aryan Gaikwad
Mar 21, 2015

int fact = 1, n = 0;
for(int i = 1; i <= 7; i++)
    fact *= i;
for(int i = 1; i <= fact; i++)
    if(fact % i == 0 && i % 2 == 1)
            n++;
System.out.println(n);

Fun! But Cheap...do the math :-)

Richard Overholt - 4 years, 8 months ago

instead of checking i is odd or not, after every iteration increase the value of i by 2

A Former Brilliant Member - 4 years, 7 months ago

Yesss! I love coding!

Zoe Codrington - 2 years, 9 months ago

O(n) is not worth it for a simple math like this. Imagine numbers like !10000, it will take minutes to calculate.

Hossain Nahdi - 2 months, 3 weeks ago

7!

The answer is 12.

3 solutions

1 pending report

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