ISEE Upper Level Quantitative Reasoning - How to factor a number

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

Explanation

The elements are as follows:

$A = \left { 5, 10, 15, 20...395 \right }$

This can be rewritten as

$A = \left { 5 \times 1, 5\times 2, ... 5 \times 79 \right }$ .

Therefore, there are elements in .

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

Explanation

The elements are as follows:

$A = \left { 3, 6, 9, 12, 15...399 \right }$

This can be rewritten as

$A = \left { 3 \times 1, 3 \times 2, ... 3 \times 133 \right }$ .

Therefore, there are elements in .

What are all of the prime factors of 34?

Explanation

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.

Add the factors of 19.

Explanation

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

What is the prime factorization of 78?

Explanation

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:

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

Explanation

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

Their sum is .

Adam fills up $\dpi{100} \frac{3}{4}$ of his glass in $\dpi{100} \frac{1}{2}$ of a minute. What is the total time, in seconds, that it takes him to fill up his entire glass?

$\dpi{100} 40\ seconds$

$\dpi{100} 60\ seconds$

$\dpi{100} 45\ seconds$

$\dpi{100} 50\ seconds$

Explanation

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

The easiest was probably involves converting the $\dpi{100} \frac{1}{2}$ minute to 30 seconds as soon as possible.

Now we can see that Adam has filled $\dpi{100} \frac{3}{4}$ of his cup in 30 seconds. We can also see that he needs to fill $\dpi{100} 1-\frac{3}{4}=\frac{1}{4}$ 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.

Add the factors of .

Explanation

The factors of are:

Their sum is .

Give the prime factorization of 135.

$3 \times 3 \times 3 \times5$

$3 \times 5\times 9$

$3 \times 3 \times 15$

$3 \times 3 \times 5 \times 5$

$3 \times 3 \times 3 \times 3 \times 5$

Explanation

$135 = 5 \times 27 = 5 \times 3 \times 9= 5 \times 3 \times 3 \times 3$

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

$135 =3 \times 3 \times 3 \times 5$ .

How many integers from 51 to 70 inclusive do not have 2, 3, or 5 as a factor?

None

Explanation

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: