ISEE Upper Level Math : How to factor a number

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

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Example Question #1 : How To Factor A Number

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

Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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 : Other Factors / Multiples

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

Possible Answers:

Correct answer:

Explanation:

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

Their sum is .

Example Question #4 : Other Factors / Multiples

Give the prime factorization of 135.

Possible Answers:

Correct answer:

Explanation:

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

.

Example Question #3 : 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:

Explanation:

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 #4 : 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.

Explanation:

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 #5 : How To Factor A Number

Which of the following is divisible by ?

Possible Answers:

Correct answer:

Explanation:

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 : Other Factors / Multiples

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

Possible Answers:

Correct answer:

Explanation:

The elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

Example Question #6 : How To Factor A Number

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

Possible Answers:

Correct answer:

Explanation:

The elements are as follows:

This can be rewritten as

.

Therefore, there are elements in .

Example Question #7 : How To Factor A Number

Add the factors of 19.

Possible Answers:

Correct answer:

Explanation:

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

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