Use a tree to find all of the factors of .

The prime factors of are .

There are more than one ways to go about solving this problem.

The easiest was probably involves converting the minute to 30 seconds as soon as possible. 

Now we can see that Adam has filled of his cup in 30 seconds.  We can also see that he needs to fill of his cup to fill his cup entirely.  Since 3 of those quarters fill up in 30 seconds, then 1 of those quarters can be filled in 10 seconds  Thus Adam needs an additional 10 seconds to finish filling his glass, or a total of 40 seconds.

To find the prime factorization, start by breaking 72 down. I picked , which can be broken down further to . The 4 can be broken down further, so go one more step to . The answers are given in exponents so give your answer in that format:

What are all of the prime factors of 34?

We need to find which prime numbers can be multiplied to get to 34. 

We can find these numbers by dividing prime numbers out one at a time.

Recall that a prime factor is a number which is only divisible by one and itself.

When performing prime factorization on an even number, always begin by pulling out 2.

Now, we are essentially done, because 17 is also a prime number. So, the prime factors of 34 are 2 and 17.

What is the prime factorization of 78?

To find the prime factorization of a number, we need to find all the prime numbers which, when multiplied, give us our original number.

When starting with an even number, find the PF by first pulling out a two.

Next, what can we pull out of the 39? Let's try 3

Can we pull anything out of the 13? Nope!

Therefore, our answer is:

The prime numbers between 50 and 70 are 53, 59, 61, and 67. Their sum is

.

The prime numbers between 20 and 40 are 23, 29, 31, and 37.

Their sum is .

A prime number has exactly two factors, 1 and itself. We can eliminate four choices by finding other factors:

53 has only 1 and 53 as factors, so it is the only prime among the choices.

There are twenty integers from 61 to 80 inclusive. Counting composite numbers can be made easier by weeding out the primes - 61, 67, 71, 73, 79. Removal of these five primes leave fifteen composite numbers.

The correct answer is , and this can be determined in the following manner.

First, find the approximate square root of the number:

We know this because:

Therefore, we only need to consider prime numbers through

Is  evenly divisible by any of these numbers? In this case, the answer is no, therefore  is prime. Consider the case where the answer is not prime: .

We know this because:

Therefore, we need to consider the followig prime numbers:

Is  divisible by any of these numbers? In this case, the answer is yes.  is divisible by .