Mind-Warm-Ups Real Numbers

Problem 11: Given that b and a are real numbers, if the following are true, what is the value of a ?
   b x a = b    b/a = b
Ans: 1.

Problem 12: Given that a, b and c are real numbers, express the commutative, associative and distributive laws with respect to addition and multiplication.
Ans: a + b = b + a; ab = ba, commutative law.
a + (b + c) = (a + b) + c; a(bc) = b(ab), associative law
a(b + c) = ab + ac, distributive law.
.

Problem 13: If b is a multiple of a, which of the following is true?
   (i) b = ka, where k is an integer
   (ii) b > 1
   (iii) a is a divisor of b
   (iv) a is a factor of b
   (v) all of the above
Ans: all of the above.

Problem 14: Given that b is an even number and a is an odd number, express b and a in terms of an arbitrary integer.
Ans: b = 2n; a = 2n + 1, where n is an arbitrary integer.

Problem 15: Find a real number x for which the nearest greatest integer ≤ x = the nearest least integer ≥ x.
Ans: all x in the range n < x < n + 1, where n is an integer.

Problem 16: Given that b and a are positive 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?
   (i) ax <= b < a(x +1)
   (ii) b = ax + r
   (iii) 0 <= r < a
   (iv) all of the above
Ans: all of the above.

Problem 17: if q is the quotient of the division of b by a (b,a are integers) and r is the remainder, determine the greatest integer that is less than or equal to the ratio b/a.
Ans: q = [b/a]. In general, [x] is the greatest integer ≤ x (also called the floor function). The least integer ≥ x, is called the ceil function.

Problem 18: show that the square of an integer is either divisible by 4 or leaves the remainder 1 when divided by 4.
Ans: any integer n is either even or odd. If it is even it is of the form 2n. If it is odd, it is of the form 2n + 1. So squaring an even number we have 4n2. Squaring an odd number we have, 4n2 + 4n + 1. Therefore, the remainder is either 0 or 1 when the square of any integer is divided by 4.

Problem 19: integer c is a common divisor of integers a nd b, if c divides a and b simultaneously. Amongst the common divisor of a and b, there is the greatest common divisor (g.c.d). It is usually denoted as (a, b). Evaluate (77,49).
Ans: 7.

Problem 20: m, an integer is a common multiple of integers a and b if m is divisible by both of them. Amongs common multiples of a and b, there is the least common multiple (l.c.m) denoted by [a, b]. Evaluate [4, 8]
Ans: 8.

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Peter Oye Sagay

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