Evaluate the following statements regarding the properties and classification of numbers. … — State PSC Exam

Evaluate the following statements regarding the properties and classification of numbers. Which of the statements are mathematically correct? (Select all that apply)

The product of the irrational numbers $\sqrt{8}$ and $\sqrt{18}$ is a composite integer.Answer
B
The unit digit of the exponential number $2^{24}$ is an odd integer.
The sum of any two distinct odd prime numbers is always an even composite integer.Answer
D
The remainder obtained when $-17$ is divided by $5$ is an even integer.

Answer

The statement regarding the product of

\sqrt{8}

and

\sqrt{18}

being a composite integer, and the statement that the sum of any two distinct odd prime numbers is an even composite integer, are both mathematically correct.

The statement about the irrational product is correct because

\sqrt{8} \times \sqrt{18} = 12

, and 12 is a composite integer. The statement regarding prime numbers is correct because adding any two odd numbers always yields an even number. Since the smallest distinct odd primes are 3 and 5 (summing to 8), their sum will always be an even integer strictly greater than 2, which guarantees it is composite.

Step-by-Step Solution

Evaluate the product of the given irrational numbers.

\sqrt{8} \times \sqrt{18} = \sqrt{144} = 12

. Since 12 has factors other than 1 and itself, it is a composite integer.

To verify if the statement about the irrational product correctly classifies the resulting number.

Calculate the unit digit of

2^{24}

The exponent 24 is perfectly divisible by 4 (

24 \pmod 4 = 0

). We use the 4th power in the cycle:

2^4 = 16

. The unit digit is 6, which is even.

To determine the parity of the unit digit for the relevant statement.

Analyze the sum of two distinct odd prime numbers.

An odd integer added to another odd integer yields an even integer. The smallest possible sum is

3 + 5 = 8

. Any even integer greater than 2 is composite.

To mathematically prove the validity of the statement concerning prime sums.

Determine the remainder when

-17

is divided by 5.

Using modulo arithmetic,

-17 = 5 \times (-4) + 3

. The true positive remainder is 3, which is an odd integer.

To correctly evaluate the division operation and classify the remainder for the modulo statement.

Key Concept

Classification of Numbers and Number Properties

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic