Tell whether each sentence is True or False.
The number 96 is a multiple of 8. That means
all of the factors of 8 are also factors of 96.
The number 1 is prime.
The number 1 is composite.
The number 2 Is prime.
The number 9 has four factors.
The Correct Answer and Explanation is :
Here are the statements along with their truth values and explanations:
- The number 96 is a multiple of 8. That means all of the factors of 8 are also factors of 96.
True.
Since 96 is a multiple of 8, we can express it as ( 96 = 8 \times 12 ). The factors of 8 are 1, 2, 4, and 8. Because 96 can be divided evenly by 8, all factors of 8 (1, 2, 4, 8) are indeed also factors of 96. - The number 1 is prime.
False.
A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. Since 1 does not meet the criteria of being greater than 1, it is not considered a prime number. - The number 1 is composite.
False.
A composite number is defined as a natural number greater than 1 that has more than two distinct positive divisors. Since 1 has only one positive divisor (itself), it is neither prime nor composite. - The number 2 is prime.
True.
The number 2 is the smallest and the only even prime number. It has exactly two distinct positive divisors: 1 and 2. - The number 9 has four factors.
False.
The factors of 9 are 1, 3, and 9. Hence, it has only three factors.
Summary
The first statement is true because being a multiple ensures that all factors of the divisor are also factors of the multiple. The statements about the number 1 being prime and composite are false; it doesn’t meet the criteria for either classification. The statement about 2 is true as it is a prime number. Lastly, the number 9 has three factors, making the claim of four factors false. This exploration of factors and primes reveals important properties of numbers, emphasizing that prime numbers are fundamental in number theory and play crucial roles in various mathematical contexts, including cryptography and factorization.