### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #1 : How To Factor A Number

What is the product of all of the factors of 25?

**Possible Answers:**

**Correct answer:**

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

### Example Question #2 : How To Factor A Number

Which of these numbers has exactly three factors?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #3 : How To Factor A Number

What is the sum of all of the factors of 60?

**Possible Answers:**

**Correct answer:**

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

Their sum is .

### Example Question #4 : How To Factor A Number

Give the prime factorization of 135.

**Possible Answers:**

**Correct answer:**

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

.

### Example Question #5 : How To Factor A Number

Which of the following digits can go into the box to form a three-digit number divisible by 3?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #6 : How To Factor A Number

Which of the following digits can go into the box to form a three-digit number divisible by 4?

**Possible Answers:**

None of the other choices is correct.

**Correct answer:**

None of the other choices is correct.

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.

### Example Question #7 : How To Factor A Number

Which of the following is divisible by ?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #8 : How To Factor A Number

Let be the set of all integers such that is divisible by and . How many elements are in ?

**Possible Answers:**

**Correct answer:**

The elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

### Example Question #1 : Other Factors / Multiples

Let be the set of all integers such that is divisible by three and . How many elements are in ?

**Possible Answers:**

**Correct answer:**

The elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

### Example Question #1 : How To Factor A Number

Add the factors of 19.

**Possible Answers:**

**Correct answer:**

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

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