Numbers and Operations - ISEE Upper Level Quantitative Reasoning

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.

Of the four numbers given, 25 is only a factor of 150, since all multiples of 25 end in the digits 25, 50, 75, or 00. To determine whether 150 is correct, we inspect the factors of 150 and 175:

Factors of 150:

Factors of 175:

Since 25 is the greatest number in both lists, .

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 .

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.

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:

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:

: 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:

:

: None

: None

: None

Taking these together, you get:

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 .

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.

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.

27 has four factors:

Their sum is .

Both are prime factors so this is the prime factorization.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is

.

40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.

(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.

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.

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:

(a) The factors of are Their sum is

.

(b) The factors of are Their sum is

.

(b) is greater.

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: