Problem number eleven states, what proportion of all divisors of seven factorial is odd? So first let's figure out how to know how many divisors a number has. So let's say that we have a number n with a prime factorization of p to the a times q to the b times r to the c and so on, where p, q and r are prime numbers and a, b and c are natural numbers. The number of divisors that n has is going to be a plus one times b plus one times c plus one and so on for all the powers of the prime factors. So here's an example, twenty can be broken up into two squared times five. So if we calculate the number of divisors it has, we get two plus one times one plus one, so three times two which is six. So now let's take a look at seven factorial. Seven factorial is just seven times six times five times four times three times two times one. So let's break up each of the terms into their prime factorizations and let's collect the terms. So seven factorial has one plus one times one plus one times two plus one times four plus one divisors, which works out to sixty. So now we know how many divisors we have, but how do we know how many of them are odd? An odd product can only happen if the numbers multiplying it are all odd. So seven times three will be an odd number, but seven times four or eight times four will not be. So let's take a look at seven factorial again. And now let's cut out all of the even terms. Let's recalculate the number of divisors now, so one plus one times one plus one times one plus one. So that's going to evaluate to twelve. So we have sixty divisors and twelve of them are odd. So that's a proportion of twelve to sixty, or one in five. So the question asked us, what proportion of all divisors of seven factorial is odd? The answer is one in five, letter D.

odd divisors

7 factorial

prime factorization

proportion

mathematics