Counting - SAT Math
Card 0 of 7
How many prime numbers are between 1 to 25?
How many prime numbers are between 1 to 25?
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, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Prime numbers are underlined. Nine prime numbers are in this set interval.
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, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Prime numbers are underlined. Nine prime numbers are in this set interval.
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Four consecutive odd integers sum to 40. How many of these numbers are prime?
Four consecutive odd integers sum to 40. How many of these numbers are prime?
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.
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.
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A custom-made ruler is 
long and for every 
there's a tick mark. How many tick marks are there on the ruler?
A custom-made ruler is
There will be 15 gaps of 
long but 16 tick marks because there will be a tick mark on each end of the ruler.
There will be 15 gaps of
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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 
?
The positive integer
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, 
.
We know that the remainder,
If
If
If
Thus,
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How many integers lie between 
and 
, including the end points?
How many integers lie between
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,
.
Therefore, our answer is 
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,
Therefore, our answer is
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How many integers are between 
and 
, inclusive?
How many integers are between
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.
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.
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Mrs. Lovell assigns her reading class to read pages three through sixty-four of their textbook. How many pages must each student read?
Mrs. Lovell assigns her reading class to read pages three through sixty-four of their textbook. How many pages must each student read?
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 
To count the number of pages in between these two page numbers, we want to subtract
Therefore, our answer is
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