SAT Math Help » Arithmetic » Integers » Counting » 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.

You are watching: Sum of integers from 1 to 40 inclusive

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.

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,

.

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,

.

Then, adding 1,

.

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Therefore, our answer is

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.

Therefore, our answer is

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).

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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.