Integers - SAT Math
Card 0 of 20
If x represents an even integer, which of the following expressions represents an odd integer?
If x represents an even integer, which of the following expressions represents an odd integer?
Pick any even integer (2, 4, 6, etc.) to represent x. The only value that is odd is 3_x_ + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3_x_ + 1 is the only choice that adds an odd number to the product.
Pick any even integer (2, 4, 6, etc.) to represent x. The only value that is odd is 3_x_ + 1. Any number multiplied by an even integer will be even. When an even number is added and subtracted to that product, the result will be even as well. 3_x_ + 1 is the only choice that adds an odd number to the product.
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Add: 
Add:
Add the ones digit.
Since the number is 10 or greater, the tens digit is the carryover.
Add the tens digit with the carryover.
Add the hundreds digit with the carryover.
Combine all the ones digits in our calculations.
The answer is 
.
Add the ones digit.
Since the number is 10 or greater, the tens digit is the carryover.
Add the tens digit with the carryover.
Add the hundreds digit with the carryover.
Combine all the ones digits in our calculations.
The answer is
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Find the values of the variables 
and 
in the following sequence:
Find the values of the variables
Terms are obtained by alternately adding 5 and multiplying by 3:
The missing two elements are found by multiplying by 3, then by adding 5:
Multiplying by 3 confirms that this is correct:
Terms are obtained by alternately adding 5 and multiplying by 3:
The missing two elements are found by multiplying by 3, then by adding 5:
Multiplying by 3 confirms that this is correct:
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My sister invited me to play an online word game. In the game vowels (a,e,i,o,u) are worth 3 points and consonants are worth 5. How much would I score if I use the word “University” ?
My sister invited me to play an online word game. In the game vowels (a,e,i,o,u) are worth 3 points and consonants are worth 5. How much would I score if I use the word “University” ?
In the word we have 4 vowels (3 x 4 = 12 points) and 6 consonants (5 x 6= 30). If we add the points together we get a total of 42 points.
In the word we have 4 vowels (3 x 4 = 12 points) and 6 consonants (5 x 6= 30). If we add the points together we get a total of 42 points.
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What is the sum of multiples of 10 from 10 to 140 inclusive?
What is the sum of multiples of 10 from 10 to 140 inclusive?
Listing them all, 10-20-30-40-50-60-70-80-90-100-110-120-130-140 you see you can divide the numbers in half (7 pairs). Alternatively you can take (140-10+10)/2/10, adding that additional +10 in the numerator because it is inclusive, giving you 7. Just adding the top and bottom numbers gives you 10+140 for 150. 150*7 is 1050.
Listing them all, 10-20-30-40-50-60-70-80-90-100-110-120-130-140 you see you can divide the numbers in half (7 pairs). Alternatively you can take (140-10+10)/2/10, adding that additional +10 in the numerator because it is inclusive, giving you 7. Just adding the top and bottom numbers gives you 10+140 for 150. 150*7 is 1050.
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Examine the sequence:
Give the number that replaces the square.
Examine the sequence:
Give the number that replaces the square.
After the first term, the next four terms are obtained by adding, multiplying, subtracting, and dividing by 2, in order. The four terms after that are obtained by carrying out the same operations with 4. The next term is obtained by adding 6, so the operations can be expected to be repeated with 6, the next even number.
Observe the operations as carried out with 2:
Carrying out the same steps, in order, with 4:
Carrying out the same steps, in order, with 6:
, the number that replaces the circle;
, the number that replaces the square.
After the first term, the next four terms are obtained by adding, multiplying, subtracting, and dividing by 2, in order. The four terms after that are obtained by carrying out the same operations with 4. The next term is obtained by adding 6, so the operations can be expected to be repeated with 6, the next even number.
Observe the operations as carried out with 2:
Carrying out the same steps, in order, with 4:
Carrying out the same steps, in order, with 6:
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The sum of three consecutive even numbers is 42. What is the smallest even number in this sequence?
The sum of three consecutive even numbers is 42. What is the smallest even number in this sequence?
To solve this problem. First set up a mathematical equation that represents this scenario.
"The sum of three consecutive even numbers is 42."
Let
and recalling that each consecutive even number is two values greater that the previous even number, the mathematical equation becomes
From here isolate the variable on one side of the equations with all other constants on the other side.
Subtract six from both sides.
Now divide by three.
Since 
represents the beginning number in this sequence it also represents the smallest even number in this sequence. Therefore the smallest even number is 12.
To solve this problem. First set up a mathematical equation that represents this scenario.
"The sum of three consecutive even numbers is 42."
Let
and recalling that each consecutive even number is two values greater that the previous even number, the mathematical equation becomes
From here isolate the variable on one side of the equations with all other constants on the other side.
Subtract six from both sides.
Now divide by three.
Since
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a, b, c are integers.
abc < 0
ab > 0
bc > 0
Which of the following must be true?
a, b, c are integers.
abc < 0
ab > 0
bc > 0
Which of the following must be true?
Let's reductively consider what this data tells us.
Consider each group (a,b,c) as a group of signs.
From abc < 0, we know that the following are possible:
(–, +, +), (+, –, +), (+, +, –), (–, –, –)
From ab > 0, we know that we must eliminate (–, +, +) and (+, –, +)
From bc > 0, we know that we must eliminate (+, +, –)
Therefore, any of our answers must hold for (–, –, –)
This eliminates immediately a > 0, b > 0
Likewise, it eliminates a – b > 0 because we do not know the relative sizes of a and b. This could therefore be positive or negative.
Finally, ac is a product of negatives and is therefore positive. Hence ac < 0 does not hold.
We are left with a + b < 0, which is true, for two negatives added must be negative.
Let's reductively consider what this data tells us.
Consider each group (a,b,c) as a group of signs.
From abc < 0, we know that the following are possible:
(–, +, +), (+, –, +), (+, +, –), (–, –, –)
From ab > 0, we know that we must eliminate (–, +, +) and (+, –, +)
From bc > 0, we know that we must eliminate (+, +, –)
Therefore, any of our answers must hold for (–, –, –)
This eliminates immediately a > 0, b > 0
Likewise, it eliminates a – b > 0 because we do not know the relative sizes of a and b. This could therefore be positive or negative.
Finally, ac is a product of negatives and is therefore positive. Hence ac < 0 does not hold.
We are left with a + b < 0, which is true, for two negatives added must be negative.
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Add: 
Add:
Anytime a negative number is added, it is similar to subtraction. Recovert the expression to the correct form. Simplify.
The correct answer is 
.
Anytime a negative number is added, it is similar to subtraction. Recovert the expression to the correct form. Simplify.
The correct answer is
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Add the negative numbers: 
Add the negative numbers:
In order to add the negative numbers, we need to eliminate the double signs and the parentheses. A positive and a negative sign will result in a negative sign.
Evaluate the terms on the right.
The answer is: 
In order to add the negative numbers, we need to eliminate the double signs and the parentheses. A positive and a negative sign will result in a negative sign.
Evaluate the terms on the right.
The answer is:
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The sum of three consecutive odd integers is 93. What is the largest of the integers?
The sum of three consecutive odd integers is 93. What is the largest of the integers?
Consecutive odd integers differ by 2. If the smallest integer is x, then
x + (x + 2) + (x + 4) = 93
3x + 6 = 93
3x = 87
x = 29
The three numbers are 29, 31, and 33, the largest of which is 33.
Consecutive odd integers differ by 2. If the smallest integer is x, then
x + (x + 2) + (x + 4) = 93
3x + 6 = 93
3x = 87
x = 29
The three numbers are 29, 31, and 33, the largest of which is 33.
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Which of the following could represent the sum of 3 consecutive odd integers, given that d is one of the three?
Which of the following could represent the sum of 3 consecutive odd integers, given that d is one of the three?
If the largest of the three consecutive odd integers is d, then the three numbers are (in descending order):
d, d – 2, d – 4
This is true because consecutive odd integers always differ by two. Adding the three expressions together, we see that the sum is 3_d_ – 6.
If the largest of the three consecutive odd integers is d, then the three numbers are (in descending order):
d, d – 2, d – 4
This is true because consecutive odd integers always differ by two. Adding the three expressions together, we see that the sum is 3_d_ – 6.
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, where 
and 
are distinct positive integers. Which of the following could be values of 
and 
?
Since 
and 
must be positive, eliminate choices with negative numbers or zero. Since they must be distinct (different), eliminate choices where 
. This leaves 4 and 5 (which is the only choice that does not add to 20), and the correct answer, 5 and 15.
Since
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Solve: 
Solve:
Add the ones digits:
Since there is no tens digit to carry over, proceed to add the tens digits:
The answer is 
.
Add the ones digits:
Since there is no tens digit to carry over, proceed to add the tens digits:
The answer is
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At a certain high school, everyone must take either Latin or Greek. There are 
more students taking Latin than there are students taking Greek. If there are 
students taking Greek, how many total students are there?
At a certain high school, everyone must take either Latin or Greek. There are
If there are 
students taking Greek, then there are 
or 
students taking Latin. However, the question asks how many total students there are in the school, so you must add these two values together to get:
or 
total students.
If there are
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Add: 
Add:
Add the ones digit.
Since the there is a tens digit, use that as the carryover to the next term.
Add the tens digit including the carryover.
The hundreds digit is 7.
Combine the ones digit of each calculation in order.
The answer is: 
Add the ones digit.
Since the there is a tens digit, use that as the carryover to the next term.
Add the tens digit including the carryover.
The hundreds digit is 7.
Combine the ones digit of each calculation in order.
The answer is:
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Add: 
Add:
Add the ones digit.
Carry over the one from the tens digit to the next number.
Add the tens digit with the carry over.
Carry over the one from the tens digit to the hundreds digit.
Add the hundreds digit with the carry over.
The thousands digit has no carry over. The second number has no thousands digit. This means that the thousands is one. Combine all the ones digits from each of the previous calculations.
The correct answer is: 
Add the ones digit.
Carry over the one from the tens digit to the next number.
Add the tens digit with the carry over.
Carry over the one from the tens digit to the hundreds digit.
Add the hundreds digit with the carry over.
The thousands digit has no carry over. The second number has no thousands digit. This means that the thousands is one. Combine all the ones digits from each of the previous calculations.
The correct answer is:
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If m and n are both even integers, which of the following must be true?
l. _m_2/_n_2 is even
ll. _m_2/_n_2 is odd
lll. _m_2 + _n_2 is divisible by four
If m and n are both even integers, which of the following must be true?
l. _m_2/_n_2 is even
ll. _m_2/_n_2 is odd
lll. _m_2 + _n_2 is divisible by four
While I & II can be true, examples can be found that show they are not always true (for example, 22/22 is odd and 42/22 is even).
III is always true – a square even number is always divisible by four, and the distributive property tell us that adding two numbers with a common factor gives a sum that also has that factor.
While I & II can be true, examples can be found that show they are not always true (for example, 22/22 is odd and 42/22 is even).
III is always true – a square even number is always divisible by four, and the distributive property tell us that adding two numbers with a common factor gives a sum that also has that factor.
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Let S be a set that consists entirely of even integers, and let T be the set that consists of each of the elements in S increased by two. Which of the following must be even?
I. the mean of T
II. the median of T
III. the range of T
Let S be a set that consists entirely of even integers, and let T be the set that consists of each of the elements in S increased by two. Which of the following must be even?
I. the mean of T
II. the median of T
III. the range of T
S consists of all even integers. If we were to increase each of these even numbers by 2, then we would get another set of even numbers, because adding 2 to an even number yields an even number. In other words, T also consists entirely of even numbers.
In order to find the mean of T, we would need to add up all of the elements in T and then divide by however many numbers are in T. If we were to add up all of the elements of T, we would get an even number, because adding even numbers always gives another even number. However, even though the sum of the elements in T must be even, if the number of elements in T was an even number, it's possible that dividing the sum by the number of elements of T would be an odd number.
For example, let's assume T consists of the numbers 2, 4, 6, and 8. If we were to add up all of the elements of T, we would get 20. We would then divide this by the number of elements in T, which in this case is 4. The mean of T would thus be 20/4 = 5, which is an odd number. Therefore, the mean of T doesn't have to be an even number.
Next, let's analyze the median of T. Again, let's pretend that T consists of an even number of integers. In this case, we would need to find the average of the middle two numbers, which means we would add the two numbers, which gives us an even number, and then we would divide by two, which is another even number. The average of two even numbers doesn't have to be an even number, because dividing an even number by an even number can produce an odd number.
For example, let's pretend T consists of the numbers 2, 4, 6, and 8. The median of T would thus be the average of 4 and 6. The average of 4 and 6 is (4+6)/2 = 5, which is an odd number. Therefore, the median of T doesn't have to be an even number.
Finally, let's examine the range of T. The range is the difference between the smallest and the largest numbers in T, which both must be even. If we subtract an even number from another even number, we will always get an even number. Thus, the range of T must be an even number.
Of choices I, II, and III, only III must be true.
The answer is III only.
S consists of all even integers. If we were to increase each of these even numbers by 2, then we would get another set of even numbers, because adding 2 to an even number yields an even number. In other words, T also consists entirely of even numbers.
In order to find the mean of T, we would need to add up all of the elements in T and then divide by however many numbers are in T. If we were to add up all of the elements of T, we would get an even number, because adding even numbers always gives another even number. However, even though the sum of the elements in T must be even, if the number of elements in T was an even number, it's possible that dividing the sum by the number of elements of T would be an odd number.
For example, let's assume T consists of the numbers 2, 4, 6, and 8. If we were to add up all of the elements of T, we would get 20. We would then divide this by the number of elements in T, which in this case is 4. The mean of T would thus be 20/4 = 5, which is an odd number. Therefore, the mean of T doesn't have to be an even number.
Next, let's analyze the median of T. Again, let's pretend that T consists of an even number of integers. In this case, we would need to find the average of the middle two numbers, which means we would add the two numbers, which gives us an even number, and then we would divide by two, which is another even number. The average of two even numbers doesn't have to be an even number, because dividing an even number by an even number can produce an odd number.
For example, let's pretend T consists of the numbers 2, 4, 6, and 8. The median of T would thus be the average of 4 and 6. The average of 4 and 6 is (4+6)/2 = 5, which is an odd number. Therefore, the median of T doesn't have to be an even number.
Finally, let's examine the range of T. The range is the difference between the smallest and the largest numbers in T, which both must be even. If we subtract an even number from another even number, we will always get an even number. Thus, the range of T must be an even number.
Of choices I, II, and III, only III must be true.
The answer is III only.
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Divide: 
Divide:
Take a known common factor of two and rewrite the fraction.
Dividing the number 143 into 16, the coefficient is 8 since:
There is a remainder of fifteen, which is 
.
Combining the coefficient and the remainder as a mixed fraction, this can be rewritten as: 
The answer is: 
Take a known common factor of two and rewrite the fraction.
Dividing the number 143 into 16, the coefficient is 8 since:
There is a remainder of fifteen, which is
Combining the coefficient and the remainder as a mixed fraction, this can be rewritten as:
The answer is:
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