List and explain the following number concepts: natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, composite numbers, multiples, and factors. Provide exampl

Here are the solutions to the questions:

PART B: Short Answer Questions

11. The smallest natural number is 1.

12. Is 0 a whole number? Yes.

13. One example of a negative integer is -1. (Other examples include -2, -5, etc.)

14. Is $\sqrt{9}$ a rational number? $\sqrt{9} = 3$. Since 3 can be written as $\frac{3}{1}$, it is a rational number. Yes.

15. One irrational number between 1 and 2 is $\sqrt{2}$. (Approximately 1.414)

16. The prime numbers less than 10 are 2, 3, 5, 7.

17. Is 49 a prime or composite number? 49 can be divided by 7 ($49 = 7 \times 7$). Since it has factors other than 1 and itself, it is a composite number. Composite.

18. A prime number has exactly two factors (1 and itself).

19. The smallest multiple of 10 greater than 50 is 60.

20. A number that is a multiple of both 4 and 6 is 12. (Other examples include 24, 36, etc.)

PART C: Calculations and Explanations

21. To list all the factors of 36, we find all pairs of numbers that multiply to 36: $1 \times 36 = 36$ $2 \times 18 = 36$ $3 \times 12 = 36$ $4 \times 9 = 36$ $6 \times 6 = 36$ The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

22. To write the prime factorization of 180, we can use the division method:

    $$180 \div 2 = 90$$ $$90 \div 2 = 45$$ $$45 \div 3 = 15$$ $$15 \div 3 = 5$$ $$5 \div 5 = 1$$

    The prime factorization of 180 is $2 \times 2 \times 3 \times 3 \times 5$. Using exponents, this is $\boxed{\text{2}^2 \times \text{3}^2 \times \text{5}}$.

23. From the list: 4, 9, 11, 15, 23, 25, 29 a) Identify all prime numbers. Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves. The prime numbers are 11, 23, 29.

    b) Identify all composite numbers.
    Composite numbers are numbers greater than 1 that have more than two factors.
    The composite numbers are 4, 9, 15, 25.

24. a) Explain why 0 is a rational number. Write it as a fraction p/q. A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q$ is not zero. Zero can be written as $\frac{0}{1}$ (or $\frac{0}{2}$, $\frac{0}{3}$, etc.), where 0 is an integer and 1 is a non-zero integer. Therefore, 0 is a rational number. As a fraction $\frac{p}{q}$: $\boxed{\frac{0}{1}}$.

25. a) What is the smallest positive number that is a multiple of both 8 and 12? This is the Least Common Multiple (LCM) of 8 and 12. The smallest positive number that is a multiple of both 8 and 12 is 24.

    b) Show how you found it (e.g., method used).
    Method: Listing multiples
    Multiples of 8: 8, 16, 24, 32, ...
    Multiples of 12: 12, 24, 36, ...
    The smallest common multiple is 24.