**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 4n ^{2}. Squaring an odd number we have, 4n^{2} + 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. **

Peter Oye Sagay