| Mind-Warm Up Of The Week : Let x generate the cyclic group G. If x = 1 and the order of G is 7, what is the product of the elements of G. |
| Problem 0.1: What number is at the mid-point of the real number line? Problem 0. 2: what numbers are at the end points of the real number line? Problem 0.3: What is the general name for the numbers to the right of the mid-point of the real number line? Problem 0.4: what is the general name for the numbers to the left of the mid-point of the real number line? Problem 0.5: Classify all the numbers of the real number line into two primary groups. indicate a member of each group. Problem 0.6: One of the groups in problem 0.5 can be classified into two groups. Indicate the group and the two groups into which it can be classified. Problem 0.7: what ranges on the real number line represent fractions? Problem 0.8: If n is an arbitrary whole number, what is the whole number immediately before n and what is the whole number immediately after n? Problem 0.9: Classify the four basic arithmetic operations into two arithmetic operations. Problem 0.10: Given that a and b are numbers, if the following are true , what is the value of a? b + a =b a - b = -b b x a = 0 b/a is undefined. |
| Problem 0.11: Given that b and a are numbers, if the following are true, what is the value of a ? b x a = b b/a = b Problem 0.12: Using arbitrary numbers express the commutative, associative and distributive laws with respect to arithmetic operations. Problem 0.13: If b is a multiple of a, which of the following is true? (0.13.0): b > 1 (0.13.1): b/a = x, where x is an integer (0.13.2): a is the divisor (0.13.3): x is the quotient (0.13.4): a is an integer (0.13.5): all of the above Problem 0. 14: Given that b is an even number and a is an odd number, express b and a in terms of an arbitrary integer. Problem 0.15: Any non -empty set of positive integers contains a least integer. Describe a non-empty set wherein the least and largest element of the set have the same value. Problem 0.16: The principle of mathematical induction states that if a statement about a postive x is true for x = 0, and its truth for all x < n implies its truth for x = n, then it is true for all x. Using the axiom of problem 0.15, prove the principle of mathematical induction. Problem 0.17: Use induction to show that 3/3-n + 32/3-n is a multiple of 12 for n <= 0. Problem 0.18: Given that b and a are integers and ax is the largest multiple of a which is <= b. If r is the remainder of the division of b by a, which of the following is true? (0.18.0) ax <= b < a(x +1) (0.18.1) b = ax + r (0.18.2) 0 <= r < a (0.18.3) all of the above |
| Problem 0.19:Given that X is a non-empty set of rational numbers that contains x and y. If X is closed under multiplication and z is the product of x and y, which of the following is true (0.19.0) z is a rational number (0.19.1) z = xy (0.19.2) z is a fraction (0.19.3) all of the above Problem 1.20: Given that X is a non-empty set of integers that is closed under subtraction, if x and y are members of X, show that X is also closed under addition. Problem 0.21: Show that the sum of two odd numbers is an even number. Problem 0.22: What is the least common divisor (l.c.d) of 4, 8 and 12? What is the greatest common divisor (g.c.d) of 4, 8 and 12. Problem 0.23: Suppose x and y are integers, x >1 and y >1. The integer x is a prime number if the only integers that divide it without a remainder are itself and 1. Integers x and y are mutually relatively prime if the greatest common divisor is 1 ( x is said to be prime to y and y prime to x). Express the integer 12 as a sum of two mutually relatively prime integers Problem 0. 24: All prime numbers are odd numbers except one. Which even number is a prime number? Problem 0.25: Every integer > 1 can be written as a product of primes. Express 2500 and 64 as products of primes. Problem 0.26: A prime divisor of the product of two or more integers divides at least one of the factors of the product. Which of the factors of 63 is a multiple of its prime divisor, 3. Problem 0. 27: Prove that there are infinitely many primes. Problem 0.28: The number 17 is from the decimal number system. Write 17 in the binary and hexadecimal number systems. Problem 0.29: Show that the sum of two odd numbers is an even number. |
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