Parities of number of divisors?

Number Theory Level 3

I like to count the number of positive divisors of positive integers (inclusive of 1 and itself).

In my scrapbook, I've written the number of positive divisors of the first 2500 positive integers.

If $X\%$ of these 2500 numbers are odd numbers, what is $X$ ?

The answer is 2.

2 solutions

David Vreken
Sep 17, 2018

Most integers have an even number of divisors, since each divisor is multiplied by another divisor to give the integer, except when the two divisors being multiplied are the same, which happens for square numbers. (For example, the divisors of the non-square integer $8$ are ( $1$ and $8$ ) and ( $2$ and $4$ ) for an even number of divisors, but the divisors of the square integer $9$ are ( $1$ and $9$ ) and $3$ for an odd number of divisors.)

Therefore, the only integers with an odd number of divisors are the square integers. Since $50^2 = 2500$ , there are $50$ square numbers in the first $2500$ positive integers, and therefore $50$ integers with an odd number of divisors, which is $\frac{50}{2500} = \frac{1}{50} = \boxed{2}\%$ of the total.

By the way, a fun math class activity related to this question is to give each student a number and as the teacher calls out the positive integers in increasing order (1, 2, 3, etc.) the student alternates standing up and sitting down if the number that is called out is a divisor of their number. If done correctly, the students with square numbers will be left standing (because they have an odd number of divisors) and the students with non-square numbers will be sitting (because they have an even number of divisors).

David Vreken - 2 years, 8 months ago

This brings a tear to my eye, I don't even remember when was my last "fun math class activity."

Pi Han Goh - 2 years, 8 months ago

Pradeep Tripathi
Sep 17, 2018

Answer is 2.96

No, the answer is not equal to 2.96.

Pi Han Goh - 2 years, 8 months ago

Parities of number of divisors?

The answer is 2.

2 solutions

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