# Number Properties II

The ORDER OF OPERATIONS is important when doing problems on the ACT: Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction.

 The test writer wants to know if you can do problems in the right order. You will see both arithmetic and algebra questions that test your skills at doing the math steps in the correct sequence. Sometimes they look like algebra but you simply plug in the given numbers and treat it as an arithmetic question using: “Please Excuse My Dear Aunt Sally” or “PEMDAS”. Watch the signs!

What is the value of (-x)3 + (x + y) when x = -2 and y = -3?

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The FACTORS of an integer are the integers that divide evenly into it without having a remainder. Every integer (other than 1) has at least two different factors: 1 and itself.

 The test writer wants to know if you can determine all the factors of a given integer.

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A GREATEST COMMON FACTOR (GCF) is the largest factor that is shared by two integers.

 The test writer wants to know if you recognize what a factor is and are able to identify the largest factor that is shared by two different numbers.

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The MULTIPLES of an integer are the numbers that it divides into evenly. It’s the opposite of factors.

 The test writer wants to know if you realize that every integer has an infinite number of multiples.

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The LEAST COMMON MULTIPLE (LCM) is the smallest multiple that is shared by two integers.

 The test writer wants to know if you recognize what an LCM is. For example, what is the LCM of 2 and 3?

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A PRIME number is an integer that has exactly two different factors: 1 and itself.

 The test writer wants to know if you realize that 2 is the smallest and only even prime number. 1 is NOT a prime number. (It doesn’t have two DIFFERENT factors.)

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RELATIVE PRIMES are two integers that have no prime factors in common. For example, the prime factors of 12 are 2 and 3 while the prime factors of 77 are 7 and 11. Since 12 and 77 don’t have any common prime factors, they are “relative primes.”

 The test writer wants to know if you understand this concept. There’s nothing tricky about the questions.

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The REMAINDER is what’s left over if a division problem does not come out evenly.

 The test writer wants to know if you realize the decimals on your calculator when you divide are not the remainder. In the test writer’s mind, the remainder is an integer that is represented by the numerator of a fraction with the denominator being the number you divided by.

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A RECIPROCAL is a fraction or integer turned upside down..

 The test writer want to know if you know the reciprocal of a simple fraction is a case when the numerator and denominator are reversed. For example, 2/3 becomes 3/2. For an integer, treat it as a fraction with a denominator of 1. For example, 4/1 becomes 1/4. The tricky ones are numbers with decimals, for example, 2.5. Treat it as 2.5/1, flip it over 1/2.5, multiply by 10/10 to clear the decimal and you get 10/25 which is .4.

## Sample Questions

1.  How many prime factors does the number 24 have?

A. 6
B. 4
C. 3
D. 2
E. 0

2.  A pair of numbers have a greatest common factor of 3 and a least common multiple of 18. What could be the two numbers?

A. 3 and 6
B. 3 and 12
C. 6 and 9
D. 6 and 12
E. 9 and 12

3.  In a division problem , the remainder is r. How many possible odd values for r exist?

A. 0
B. 1
C. 2
D. 3
E. 6

4.  The following formula is used by the ACT test writers to calculate the cost (C) of developing a new test using the daily wage (w), the number of days (d) it takes to write the test adjusted for absences (a), the number of questions (n), and the printing costs (P): What would the cost of a test be in dollars if: the test writers are paid \$200 per day; it takes 400 days; workers are absent 20 days; there are 200 questions to write; and printing costs \$10,000?

A. 134,400
B. 134,410
C. 136,410
D. 134,410,000
E. 136,400,000

5. A. -18
B. -12
C. 6
D. 12
E. 18

6.  Two integers have a greatest prime factor of 3 and a least common multiple of 36. Which of the following could be the integers?

A. 1 and 12
B. 3 and 18
C. 3 and 36
D. 6 and 24
E. 12 and 72

7.  Which of the following is the largest fraction?

A. 5/16
B. 7/9
C. 19/24
D. 7/12
E. 17/36

8.  Which of the following is the largest number?

A. B. C. D. E. 9.  What is the value of A. 8
B. 10
C. 12
D. 16
E. 24

10. A. 6
B. 10
C. 18
D. 24
E. 32

11.  Which of the following pairs of numbers are relative primes?

A. 12 and 18
B. 10 and 5
C. 8 and 2
D. 6 and 4
E. 3 and 2