Other Factors / Multiples - ISEE Upper Level Quantitative Reasoning

3 and 5 are both primes, so this is as far as we can go. Rearranging, the prime factorization is

.

60 has twelve factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Their sum is .

25 has three factors: 1, 5, and 25. Their product is

None of the choices are prime, so each has at least three factors. The question, then, is which one has only three factors?

We can eliminate four choices by showing that each has at least four factors - that is, at least two different factors other than 1 and itself:

Each, therefore, has at least four factors.

However, the only way to factor 121 other than is . Therefore, 121 has only 1, 11, and 121 as factors, and it is the correct choice.

Place each of these digits into the box in turn. Divide each of the numbers formed and see which quotient yields a zero remainder:

Only 627 is divisible by 3 so the correct choice is 2.

For a number to be divisible by 4, the last two digits must form an integer divisible by 4. 2 (02), 22, 62, and 82 all yield remainders of 2 when divided by 4, so none of these alternatives make the number a multiple of 4.

Numbers that are divisble by 6 are also divisble by 2 and 3. Only even numbers are divisible by 2, therefore, 72165 is excluded. The sum of the digits of numbers divisible by 3 are also divisible by 3. For example,

Because 18 is divisible by 3, 63,072 is divisible by 3.

The factors of are:

Their sum is .

19 is a prime number and has 1 and 19 as its only factors. Their sum is 20.

Use a tree to find all of the factors of .

The prime factors of are .

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 elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

The elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

We can eliminate the ten even integers right off the bat, since, by definition, all have as a factor. Of the remaining (odd) integers, we eliminate and , as they have as a factor. What remains is:

We can now eliminate the multiples of . This leaves

.

The correct choice is .

25 has three factors: 1, 5, and 25. Their product is

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

The easiest was probably involves converting the $\frac{1}{2}$ minute to 30 seconds as soon as possible.

Now we can see that Adam has filled $\frac{3}{4}$ of his cup in 30 seconds. We can also see that he needs to fill 1-$\frac{3}{4}$=\frac{1}{4}$ 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.

3 and 5 are both primes, so this is as far as we can go. Rearranging, the prime factorization is

.