Praxis Math Number Theory and Mathematical Proof Quiz

1. What is the greatest common divisor (GCD) of 48 and 18?

6

12

18

24

To find the GCD of 48 and 18, we can list the divisors of each number. The divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48, while the divisors of 18 are 1, 2, 3, 6, 9, and 18. The largest common divisor is 6.

Explanation

To find the GCD of 48 and 18, we can list the divisors of each number. The divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48, while the divisors of 18 are 1, 2, 3, 6, 9, and 18. The largest common divisor is 6.

2. Which of the following is a prime number?

91

97

99

100

97 is a prime number because it has no positive divisors other than 1 and itself. Unlike 91, 99, and 100, which can be divided evenly by other numbers, 97 cannot be divided evenly by any integers other than 1 and 97, confirming its status as a prime number.

Explanation

97 is a prime number because it has no positive divisors other than 1 and itself. Unlike 91, 99, and 100, which can be divided evenly by other numbers, 97 cannot be divided evenly by any integers other than 1 and 97, confirming its status as a prime number.

3. If a ≡ 5 (mod 7), which value is congruent to a?

12

19

24

26

To find a value congruent to \( a \equiv 5 \mod 7 \), we can add multiples of 7 to 5. The value 19 can be derived from \( 5 + 2 \times 7 = 19 \), confirming that 19 leaves a remainder of 5 when divided by 7, making it congruent to \( a \).

Explanation

To find a value congruent to \( a \equiv 5 \mod 7 \), we can add multiples of 7 to 5. The value 19 can be derived from \( 5 + 2 \times 7 = 19 \), confirming that 19 leaves a remainder of 5 when divided by 7, making it congruent to \( a \).

4. What is the remainder when 37 is divided by 5?

1

2

3

4

To find the remainder when 37 is divided by 5, perform the division: 37 divided by 5 equals 7 with a remainder. Multiplying 5 by 7 gives 35, and subtracting this from 37 results in 2. Therefore, the remainder is 2.

Explanation

To find the remainder when 37 is divided by 5, perform the division: 37 divided by 5 equals 7 with a remainder. Multiplying 5 by 7 gives 35, and subtracting this from 37 results in 2. Therefore, the remainder is 2.

5. Which proof method assumes the negation of the conclusion?

Direct proof

Proof by contradiction

Proof by induction

Proof by cases

Proof by contradiction assumes that the conclusion is false to demonstrate that this assumption leads to a logical inconsistency. By showing that the negation of the conclusion results in a contradiction, it confirms that the original statement must be true. This method effectively highlights the necessity of the conclusion by eliminating the possibility of its falsehood.

Explanation

Proof by contradiction assumes that the conclusion is false to demonstrate that this assumption leads to a logical inconsistency. By showing that the negation of the conclusion results in a contradiction, it confirms that the original statement must be true. This method effectively highlights the necessity of the conclusion by eliminating the possibility of its falsehood.

6. If p and q are both prime numbers greater than 2, what can be said about p + q?

It is always prime

It is always even

It is always odd

It is always composite

When both p and q are prime numbers greater than 2, they must be odd. The sum of two odd numbers is always even. However, since the only even prime is 2, and both p and q cannot be 2, their sum p + q is odd, confirming that it cannot be prime or even.

Explanation

When both p and q are prime numbers greater than 2, they must be odd. The sum of two odd numbers is always even. However, since the only even prime is 2, and both p and q cannot be 2, their sum p + q is odd, confirming that it cannot be prime or even.

7. What is the least common multiple (LCM) of 12 and 18?

24

30

36

48

To find the least common multiple (LCM) of 12 and 18, we can list their multiples. The multiples of 12 are 12, 24, 36, 48, and those of 18 are 18, 36, 54. The smallest common multiple in both lists is 36, making it the LCM of 12 and 18.

Explanation

To find the least common multiple (LCM) of 12 and 18, we can list their multiples. The multiples of 12 are 12, 24, 36, 48, and those of 18 are 18, 36, 54. The smallest common multiple in both lists is 36, making it the LCM of 12 and 18.

8. In mathematical induction, what must be verified in the base case?

The formula holds for all integers

The formula holds for the initial value (usually n = 1)

The inductive step is true

The formula is divisible by n

In mathematical induction, the base case serves to establish that the statement is true for the initial value, typically n = 1. This step is crucial as it lays the foundation for the inductive step, where the assumption is made that if the statement holds for n = k, it must also hold for n = k + 1.

Explanation

In mathematical induction, the base case serves to establish that the statement is true for the initial value, typically n = 1. This step is crucial as it lays the foundation for the inductive step, where the assumption is made that if the statement holds for n = k, it must also hold for n = k + 1.

9. Which statement is true about the divisibility of 15 by 3?

15 ≡ 0 (mod 3)

15 ≡ 1 (mod 3)

15 ≡ 2 (mod 3)

15 is not divisible by 3

15 is divisible by 3 because when you divide 15 by 3, the result is 5 with no remainder. In modular arithmetic, this is expressed as 15 ≡ 0 (mod 3), indicating that 15 leaves a remainder of 0 when divided by 3, confirming its divisibility.

Explanation

15 is divisible by 3 because when you divide 15 by 3, the result is 5 with no remainder. In modular arithmetic, this is expressed as 15 ≡ 0 (mod 3), indicating that 15 leaves a remainder of 0 when divided by 3, confirming its divisibility.

10. If 7 | 42, which statement is correct?

7 is a multiple of 42

42 is a divisor of 7

7 divides 42

42 and 7 are coprime

Since 7 divides 42, it means that when 42 is divided by 7, the result is an integer (specifically, 6). This indicates that 42 can be evenly split into groups of 7, confirming that 7 is a factor of 42. Thus, the statement "7 divides 42" is accurate.

Explanation

Since 7 divides 42, it means that when 42 is divided by 7, the result is an integer (specifically, 6). This indicates that 42 can be evenly split into groups of 7, confirming that 7 is a factor of 42. Thus, the statement "7 divides 42" is accurate.

11. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a product of ____.

Every integer greater than 1 can be uniquely expressed as a product of prime numbers, which are the building blocks of all integers. This theorem highlights the importance of prime factorization in number theory, ensuring that each integer can be broken down into its prime components in only one way, disregarding the order of factors.

Explanation

Every integer greater than 1 can be uniquely expressed as a product of prime numbers, which are the building blocks of all integers. This theorem highlights the importance of prime factorization in number theory, ensuring that each integer can be broken down into its prime components in only one way, disregarding the order of factors.

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12. Two integers are coprime if their greatest common divisor is ____.

Two integers are considered coprime if their greatest common divisor (GCD) is one. This means that they share no common positive integer factors other than one, indicating that they are relatively prime to each other. For example, the integers 8 and 15 are coprime because their only common divisor is one.

Explanation

Two integers are considered coprime if their greatest common divisor (GCD) is one. This means that they share no common positive integer factors other than one, indicating that they are relatively prime to each other. For example, the integers 8 and 15 are coprime because their only common divisor is one.

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