Numbers and Operations - ISEE Upper Level: Quantitative Reasoning
Card 1 of 976
3/5 + 4/7 – 1/3 =
3/5 + 4/7 – 1/3 =
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We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35.
Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.
We need to find a common denominator to add and subtract these fractions. Let's do the addition first. The lowest common denominator of 5 and 7 is 5 * 7 = 35, so 3/5 + 4/7 = 21/35 + 20/35 = 41/35.
Now to the subtraction. The lowest common denominator of 35 and 3 is 35 * 3 = 105, so altogether, 3/5 + 4/7 – 1/3 = 41/35 – 1/3 = 123/105 – 35/105 = 88/105. This does not simplify and is therefore the correct answer.
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is an odd prime.
Which is the greater quantity?
(a) 
(b) 
Which is the greater quantity?
(a)
(b)
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The greatest common factor of two numbers is the product of the prime factors they share; if they share no prime factors, it is 
.
(a) 
. Since 
is an odd prime, 
and 
share no prime factors, and 
.
(b) 
, since 
is prime. Since 
is an even prime, 
and 
share no prime factors, and 
.
The quantities are equal since each is equal to 
.
The greatest common factor of two numbers is the product of the prime factors they share; if they share no prime factors, it is
(a)
(b)
The quantities are equal since each is equal to
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Column A Column B
The GCF of The GCF of
45 and 120 38 and 114
Column A Column B
The GCF of The GCF of
45 and 120 38 and 114
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There are a couple different ways to find the GCF of a set of numbers. Sometimes it's easiest to make a factor tree for each number. The factors that the pair of numbers have in common are then multiplied to get the GCF. So for 45, the prime factorization ends up being: 
. The prime factorization of 120 is: 
. Since they have a 5 and 3 in common, those are multiplied together to get 15 for the GCF. Repeat the same process for 38 and 114. The prime factorization of 38 is 
. The prime factorization of 114 is 
. Therefore, multiply 19 and 2 to get 38 for their GCF. Column B is greater.
There are a couple different ways to find the GCF of a set of numbers. Sometimes it's easiest to make a factor tree for each number. The factors that the pair of numbers have in common are then multiplied to get the GCF. So for 45, the prime factorization ends up being:
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Annette's family has 
jars of applesauce. In a month, they go through 
jars of apple sauce. How many jars of applesauce remain?
Annette's family has
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If Annette's family has 
jars of applesauce, and in a month, they go through 
jars of apple sauce, that means 
jars of applesauce will be left.
The first step to determining how much applesauce is left it to convert the fractions into mixed numbers. This gives us:
The next step is to find a common denominator, which would be 15. This gives us:
If Annette's family has
The first step to determining how much applesauce is left it to convert the fractions into mixed numbers. This gives us:
The next step is to find a common denominator, which would be 15. This gives us:
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What is the greatest common factor of 
and 
?
What is the greatest common factor of
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To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
: 
: 
: 
: None
: None
Taking these together, you get:
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
Taking these together, you get:
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What is the greatest common factor of 
and 
?
What is the greatest common factor of
Tap to reveal answer
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
: None
: 
: None
: None
Taking these together, you get: 
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
Taking these together, you get:
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What is the greatest common factor of 
and 
?
What is the greatest common factor of
Tap to reveal answer
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
: 
: None
: None
: None
Taking these together, you get: 
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
Taking these together, you get:
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, 
, 
, 
, and 
are five distinct prime integers. Give the greatest common factor of 
and 
.
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If two integers are broken down into their prime factorizations, their greatest common factor is the product of their common prime factors.
Since 
, 
, 
, 
, and 
are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:
The greatest common factor is the product of those three factors, or 
.
If two integers are broken down into their prime factorizations, their greatest common factor is the product of their common prime factors.
Since
The greatest common factor is the product of those three factors, or
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Which of these numbers is relatively prime with 18?
Which of these numbers is relatively prime with 18?
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For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.
We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1, 5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.
For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.
We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1, 5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.
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Which of the following is the prime factorization of 333?
Which of the following is the prime factorization of 333?
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To find the prime factorization, break the number down as a product of factors, then keep doing this until all of the factors are prime.
To find the prime factorization, break the number down as a product of factors, then keep doing this until all of the factors are prime.
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What is the sum of all of the factors of 27?
What is the sum of all of the factors of 27?
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27 has four factors: 
Their sum is 
.
27 has four factors:
Their sum is
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Give the prime factorization of 91.
Give the prime factorization of 91.
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Both are prime factors so this is the prime factorization.
Both are prime factors so this is the prime factorization.
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Add all of the factors of 30.
Add all of the factors of 30.
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The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is
.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is
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How many factors does 40 have?
How many factors does 40 have?
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40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.
40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.
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Which is the greater quantity?
(a) The number of factors of 15
(b) The number of factors of 17
Which is the greater quantity?
(a) The number of factors of 15
(b) The number of factors of 17
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(a) 15 has four factors, 1, 3, 5, and 15.
(b) 17, as a prime, has two factors, 1 and 17.
Therefore, (a) is greater.
(a) 15 has four factors, 1, 3, 5, and 15.
(b) 17, as a prime, has two factors, 1 and 17.
Therefore, (a) is greater.
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Which is the greater quantity?
(a) The number of factors of 169
(b) The number of factors of 121
Which is the greater quantity?
(a) The number of factors of 169
(b) The number of factors of 121
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Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.
Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.
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Which is the greater quantity?
(a) The product of the integers between 
and 
inclusive
(b) The sum of the integers between 
and 
inclusive
Which is the greater quantity?
(a) The product of the integers between
(b) The sum of the integers between
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The quanitites are equal, as both can be demonstrated to be equal to 
.
(a) One of the integers in the given range is , so one of the factors will be , making the product .
(b) The sum of the numbers will be:
The quanitites are equal, as both can be demonstrated to be equal to .
(a) One of the integers in the given range is , so one of the factors will be
, making the product
.
(b) The sum of the numbers will be:
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Which is the greater quantity?
(a) The sum of the factors of 
(b) The sum of the factors of 
Which is the greater quantity?
(a) The sum of the factors of
(b) The sum of the factors of
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(a) The factors of 
are 
Their sum is
.
(b) The factors of 
are 
Their sum is
.
(b) is greater.
(a) The factors of
(b) The factors of
(b) is greater.
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Which is the greater quantity?
(a) The sum of all of the two-digit even numbers
(b) 2,500
Which is the greater quantity?
(a) The sum of all of the two-digit even numbers
(b) 2,500
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The sum of the integers from 
to 
is equal to 
. We take advantage of the fact that the sum of the even numbers from 10 to 98 is equal to twice the sum of the integers from 5 to 49, as seen here:
The sum of the integers from
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What is the prime factorization of 
?
What is the prime factorization of
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First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example: 
, which then yields 
). Then, at the end, remember to factor the variables as well. Since the b term is squared, that means there are two of them. Therefore, the final answer is 
.
First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example:
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