# SAT Math : Counting

## Example Questions

### Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

A custom-made ruler is  long and for every  there's a tick mark. How many tick marks are there on the ruler?

Explanation:

There will be 15 gaps of  long but 16 tick marks because there will be a tick mark on each end of the ruler.

### Example Question #1 : Counting

How many prime numbers are between 1 to 25?

Explanation:

A prime numbers is a number greater than 1 that can only be divided by 1 and itself. Simply count from 1 to 25 and see how many values fit the criteria.

2, 3, 4, 5, 6, 78, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Prime numbers are bolded. Nine prime numbers are in this set interval.

### Example Question #1 : How To Find The Number Of Integers Between Two Other Integers

Four consecutive odd integers sum to 40. How many of these numbers are prime?

Explanation:

Let x equal the smallest of the four numbers. Therefore:

Therefore the four odd numbers are 7, 9, 11, and 13. Since all are prime except 9, three of the numbers are prime.

### Example Question #4 : How To Find The Number Of Integers Between Two Other Integers

The positive integer  is not divisible by . The remainder when  is divided by  and the remainder when  is divided by  are both equal to . What is

Explanation:

We know that the remainder, , must be less than  by the definition of remainder. Therefore our only choices are , , or . We can test each of these cases.

If   would be divisible by , which we said is not true.

If : Try . Then . When we divide  by , we have a remainder of . This works!

If : Try . Then . When we divide this by , we have a remainder of . Thus, our remainders are not equal.

Thus,

### Example Question #2 : How To Find The Number Of Integers Between Two Other Integers

How many integers lie between  and , including the end points?

Explanation:

Let's look at a small example, the numbers between 1 and 3 including all endpoints. We have 1, 2, and 3. If we subtract our numbers, we get 2. We need to add one more to get all the desired numbers. We can follow the same process with our larger numbers.

Take,

.

.

### Example Question #6 : How To Find The Number Of Integers Between Two Other Integers

How many integers are between  and

Explanation:

In order to obtain the numbers between -8 and 17, you can draw a number line starting with -8 on the left, 0 in the middle, and 17 on the right.

In order to get from -8 to 0, you must move 8 integers.

In order to get from 0 to 17, you must move 17 integers.

In total, you move  integers, which gives the final answer:  integers.

### Example Question #7 : How To Find The Number Of Integers Between Two Other Integers

Mrs. Lovell assigns her reading class to read pages three through sixty-four of their textbook. How many pages must each student read?