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Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Binary Numbers

Complex Plane

PDE

**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