# SAT Math : How to find the number of integers between two other integers

## 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 #2 : 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 #3 : 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 #4 : 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 #1 : 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?

Explanation:

To count the number of pages in between these two page numbers, we want to subtract  and then add . We have to add  because if we don't, we end up not counting either the first or last page of the selection.

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

How many integers are between  and , inclusive?

Explanation:

An important consideration whenever you're asked for the number of integers between two other numbers is to determine whether the two "end points" are included. Here the word "inclusive" tells us that we need to count both -8 and 17 in our calculation.

A helpful tip for these questions: if you're including both end points, you can calculate by using the range (greatest value minus smallest value) and adding 1 to account for the inclusive set.  (If you need to prove that to yourself, just use "how many integers between 1 and 3 inclusive" - clearly the set is 1, 2, 3 for three integers, but 3-1 = 2, so you can see you need to add one more to get the answer).

For this question, that leaves us with 17 - (8) = 25 as the range, and if you add 1 to that for "inclusive" you get the correct answer, 26.

Another way to look at this one is to chop it up: between -8 and 17 there are 8 negative integers (-1 through -8), 17 positive integers (1 through 17), and 0 as one more integer.  8 + 17 + 1 = 26.

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

How many prime numbers are between 1 to 25?