Numbers and Operations - ISEE Upper Level Quantitative Reasoning

1 raised to any exponent will always be 1.

-1 will be equal to 1 when the exponent is even and will be equal to -1 when the exponent is odd.

Given that 323 is odd, is equal to -1.

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

Find the factors of 72 and 80; the greatest common factor is the greatest number in both lists.

Fully factor and . Circle the factors in common:

Multiple the common factors to find the greatest common factor:

We are looking for a number that shares no factors with 10 other than 1. 20 and 28 can be eliminated since 2 divides 10, 20, 28. 25 and 35 can be eliminated since 5 divides 10, 25, and 25.

21 is the correct choice; and . Since they share no prime factors, , and 10 and 21 are relatively prime.

For a number to be relatively prime with , . Equivalently, and cannot share a prime factor. , so any number that does not have either or as a factor is a correct choice.

We can eliminate and , since is a factor of each; we can eliminate and , since is a factor of each. But , so . is the correct choice.

To find the greatest common factor (GCF), you need to determine the factor that both numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in both lists:

The GCF of 20 and 36 is 4 since it is the largest number in value that shows up in both lists.

To find the greatest common factor (GCF), you need to determine the factor that all three numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in all three lists:

The GCF of 15, 30, and 40 is 5 since it is the largest number in value that shows up in all three lists.

Factor each number:

Identify the common factors (). The greatest common factor is the product of all of the common factors.

The greatest common factor of two numbers is the largest shared factor that the two numbers have in common.

The factors of 36 are

The factors of 108 are

The largest shared factor between 36 and 108 is 36. Therefore, 36 is the correct answer.

Find the greatest common factor (GCF) of 56 and 64.

To find the GCF, first find the prime factors of our two numbers:

Now, to find our GCF, we need to find the prime factors our two numbers have in common.

In this case, we have three 2's in common. In other words, we have an 8 in common. So, our GCF is 8.

Find the greatest common factor (GCF) of 324 and 56.

The GCF is the largest number which will can be evenly divided from either number.

To find the GCF, first find the prime factors of both numbers. A good trick here is to start by pulling a two out of each number until you cannot pull out a two anymore.

Beginning with 56

So the prime factorization of 56 looks like:

Next, let's do 324.

So for 324 we get:

Now, the GCF can be found by taking the prime factors which are common to both numbers. Now, in this case, we have two 2's in each number, and nothing else. This means that the GCF of 56 and 324 is

So our answer is 4

Fully factor and . Circle the factors in common:

Multiple the common factors to find the greatest common factor:

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

.

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

Their sum 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.