TExES Core Mathematics Number Theory Quiz

1. Which of the following is a prime number?

91

97

99

121

A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. Among the options, 97 is the only number that meets this criterion, as it cannot be divided evenly by any other numbers except for 1 and 97.

Explanation

A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. Among the options, 97 is the only number that meets this criterion, as it cannot be divided evenly by any other numbers except for 1 and 97.

2. What is the greatest common divisor (GCD) of 48 and 180?

12

24

36

60

To find the GCD of 48 and 180, we can use the prime factorization method. The prime factors of 48 are 2^4 and 3^1, while the prime factors of 180 are 2^2, 3^2, and 5^1. The common factors are 2^2 and 3^1, giving a GCD of 2^2 × 3^1 = 12.

Explanation

To find the GCD of 48 and 180, we can use the prime factorization method. The prime factors of 48 are 2^4 and 3^1, while the prime factors of 180 are 2^2, 3^2, and 5^1. The common factors are 2^2 and 3^1, giving a GCD of 2^2 × 3^1 = 12.

3. Find the least common multiple (LCM) of 12 and 18.

36

48

54

72

To find the least common multiple (LCM) of 12 and 18, we can list their multiples or use the prime factorization method. The prime factors of 12 are 2² × 3 and for 18 are 2 × 3². The LCM is found by taking the highest power of each prime, resulting in 2² × 3² = 36.

Explanation

To find the least common multiple (LCM) of 12 and 18, we can list their multiples or use the prime factorization method. The prime factors of 12 are 2² × 3 and for 18 are 2 × 3². The LCM is found by taking the highest power of each prime, resulting in 2² × 3² = 36.

4. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

True

False

A number is divisible by 6 when it meets the criteria for both 2 and 3. Divisibility by 2 requires the number to be even, while divisibility by 3 means the sum of its digits must be divisible by 3. Therefore, satisfying both conditions ensures the number is divisible by 6.

Explanation

A number is divisible by 6 when it meets the criteria for both 2 and 3. Divisibility by 2 requires the number to be even, while divisibility by 3 means the sum of its digits must be divisible by 3. Therefore, satisfying both conditions ensures the number is divisible by 6.

5. What is the remainder when 157 is divided by 7?

2

3

4

5

To find the remainder when 157 is divided by 7, perform the division: 157 ÷ 7 = 22 with a remainder. Calculating 22 × 7 gives 154. Subtracting this from 157 results in 3. Therefore, the remainder is 3, not 5. The provided answer appears to be incorrect.

Explanation

To find the remainder when 157 is divided by 7, perform the division: 157 ÷ 7 = 22 with a remainder. Calculating 22 × 7 gives 154. Subtracting this from 157 results in 3. Therefore, the remainder is 3, not 5. The provided answer appears to be incorrect.

6. The number 2520 is the LCM of which pair?

60 and 84

120 and 210

168 and 210

180 and 280

To find the LCM of 168 and 210, we first determine their prime factors: 168 = 2^3 × 3 × 7 and 210 = 2 × 3 × 5 × 7. The LCM is calculated by taking the highest powers of all prime factors: 2^3, 3^1, 5^1, and 7^1, resulting in 2^3 × 3 × 5 × 7 = 2520.

Explanation

To find the LCM of 168 and 210, we first determine their prime factors: 168 = 2^3 × 3 × 7 and 210 = 2 × 3 × 5 × 7. The LCM is calculated by taking the highest powers of all prime factors: 2^3, 3^1, 5^1, and 7^1, resulting in 2^3 × 3 × 5 × 7 = 2520.

7. If n ≡ 5 (mod 8), which value could n be?

29

32

40

45

If \( n \equiv 5 \mod 8 \), it means that when \( n \) is divided by 8, the remainder is 5. Among the options, only 29 fits this condition, as \( 29 \div 8 \) gives a remainder of 5. The other numbers either produce different remainders or are not congruent to 5 modulo 8.

Explanation

If \( n \equiv 5 \mod 8 \), it means that when \( n \) is divided by 8, the remainder is 5. Among the options, only 29 fits this condition, as \( 29 \div 8 \) gives a remainder of 5. The other numbers either produce different remainders or are not congruent to 5 modulo 8.

8. Every integer greater than 1 can be expressed as a product of ____ numbers.

Every integer greater than 1 can be uniquely expressed as a product of prime numbers, according to the Fundamental Theorem of Arithmetic. This means that any integer can be broken down into prime factors, which are the building blocks of all integers, ensuring each number has a distinct prime factorization.

Explanation

Every integer greater than 1 can be uniquely expressed as a product of prime numbers, according to the Fundamental Theorem of Arithmetic. This means that any integer can be broken down into prime factors, which are the building blocks of all integers, ensuring each number has a distinct prime factorization.

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9. A number is divisible by 9 if the sum of its digits is divisible by 9.

True

False

A number is divisible by 9 if the sum of its digits is also divisible by 9. This rule stems from the properties of the base-10 number system, where each digit's place value can be expressed as a multiple of 9. Thus, checking the digit sum provides a quick way to determine divisibility by 9.

Explanation

A number is divisible by 9 if the sum of its digits is also divisible by 9. This rule stems from the properties of the base-10 number system, where each digit's place value can be expressed as a multiple of 9. Thus, checking the digit sum provides a quick way to determine divisibility by 9.

10. What is GCD(144, 96)?

12

24

48

72

To find the GCD (Greatest Common Divisor) of 144 and 96, we can list the prime factors of each number. The prime factorization of 144 is \(2^4 \times 3^2\) and for 96 it is \(2^5 \times 3^1\). The GCD is obtained by taking the lowest powers of the common prime factors, which gives us \(2^4 \times 3^1 = 48\).

Explanation

To find the GCD (Greatest Common Divisor) of 144 and 96, we can list the prime factors of each number. The prime factorization of 144 is \(2^4 \times 3^2\) and for 96 it is \(2^5 \times 3^1\). The GCD is obtained by taking the lowest powers of the common prime factors, which gives us \(2^4 \times 3^1 = 48\).

11. The number 1 is considered a ____ number.

The number 1 is classified as neither prime nor composite because it does not meet the criteria for either category. A prime number has exactly two distinct positive divisors (1 and itself), while a composite number has more than two. Since 1 only has one divisor, it falls outside both definitions.

Explanation

The number 1 is classified as neither prime nor composite because it does not meet the criteria for either category. A prime number has exactly two distinct positive divisors (1 and itself), while a composite number has more than two. Since 1 only has one divisor, it falls outside both definitions.

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12. Which set contains only prime numbers?

2, 3, 5, 7, 11

1, 2, 3, 5, 7

3, 5, 9, 11, 13

2, 4, 6, 8, 10

The set 2, 3, 5, 7, 11 contains only prime numbers, which are defined as natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Each number in this set meets this criterion, while the other sets include composite numbers or non-prime integers.

Explanation

The set 2, 3, 5, 7, 11 contains only prime numbers, which are defined as natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Each number in this set meets this criterion, while the other sets include composite numbers or non-prime integers.