Factors / Multiples - ISEE Upper Level Quantitative Reasoning

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.

First, we must solve for

While 64 is divisible by 4, 8, and 16, it is not divisible by 7; therefore, 7 is not a factor of 64 and is thus the correct answer.

The primes between 20 and 40 are included in both sets, so all we need to do is to compare the number of primes between 40 and 50 with the number of primes between 10 and 20.

(a) The primes between 40 and 50 are 41, 43, and 47 - three.

(b) The primes between 10 and 20 are 11, 13, 17, and 19 - four.

Since the number of primes between 10 and 20 outnumbers those between 40 and 50, the number of primes between 10 and 40 outnumber those between 20 and 50. Therefore, (b) is the greater.

The primes between 80 and 110 are included in both sets, so all we need to do is to compare the number of primes between 70 and 80 and the number of primes between 110 and 120.

(a) The primes between 70 and 80 are 71, 73, and 79 - three primes

(b) The only prime between 110 and 120 is 113.

(a) is the greater quantity

The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total. Since 1 is neither prime nor composite, this leaves 11 composite numbers. (b) is the greater quantity.

(a) The prime numbers between 1 and 20 inclusive are 2, 3, 5, 7, 11, 13, 17, 19 - eight total.

(b) The prime numbers between 21 and 30 inclusive are 23 and 29 - two prime numbers out of ten integers. This leaves eight composite numbers.

(a) and (b) are therefore equal.

(a) The prime numbers in the 11-19 range are 11, 13, 17, and 19. Their sum:

(b) The composite numbers in the 11-19 range are 12, 14, 15, 16, and 18. Their sum:

(b) is greater.

It is not necessary to count all of the composite numbers in the 1-1,000 range. Except for 2, every even number is composite, and there are more than one odd composite numbers (15, 25) to make up for 2. This makes (a) greater.

Both are prime factors so this is the prime factorization.

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 .

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

.

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:

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

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 .