**Problem 21**: Show that the sum of two odd numbers is an even number.
**Ans: an arbitrary odd number is represented as 2n + 1. So the sum of two odd numbers is 4n + 2 = 2(2n + 1), an even number.**.

**Problem 22**: What is the least common multiple (l.c.m) of 4, 8 and 12? What is the greatest common divisor (g.c.d) of 4, 8 and 12?
**Ans: g.c.d. of a and b is denoted as (a, b). g.c.d. of a, b, and c is denoted as (a, b, c) = ((a,b), c) = (a, (b,c)). So g.c.d of 4, 8, 12 =((4,8), 12) = 4.
l.c.m of a and b is denoted as [a,b]. l.c.m of a, b, and c is denoted as [[a,b],c] = [a,[b,c]]. So the l.c.m of 4, 8, 12 = [4,8],12] = 24. In general, the g.c.d and l.c.m of more than 3 numbers can be similarly determined.
l.c.m occurs frequently in the addition and subtraction of fractions when the least common denominator of the fractions are determined.**

**Problem 23**: determine the g.c.d and l.c.m of 3, 6 and 9.
**Ans: in general, the g.c.d and l.c.m of ma,mb and mc = m(a,b,c) and m[a,b,c] respectively. So (3,6,9) = 3(1,2,3) = 3. [3,6,9] = 3[1,2,3] = 18.**

**Problem 24**: An integer > 1 is a prime number if its only divisors are itself and 1. All known prime numbers are odd numbers except one. Which even number is a prime number? .
**Ans: 2. **

**Problem 25**: Every integer > 1 can be written as a product of primes. Express 2500 and 64 as products of primes.
**Ans: since √2500 = 50 and 50 is not a prime, only the primes below 50 need be examined. One method of prime factorization is to consecutively factorize the √ of the number of interest and then combine the factors as follows: 50 = 2(25) = 2(5)(5). So 2500 = 2(2)(5)(5)(5)(5) = 2 ^{2}. Simarly, 64 = 2^{6} **

**Problem 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?
**Ans: 63 = 7(9). So 9 is a multiple of the prime divisor 3. **

**Problem 27**: The factorization theorem states that every number can be represented uniquely as the product of prime numbers. Prove that there are infinitely many primes.
**Ans: let C = a, b, c, ...,k be any collection of primes numbers . The sum of the product P of these primes and 1, S = P + 1, is either a prime or not a prime. S is a prime implies a new prime can be added to any collection of primes ad infinitum, therefore there are infinitely many primes. But suppose S is not a prime. This implies S must be divisible by some prime p that is not a member of C because if it is a member of C, it would divide P and P + 1; hence it will divide 1, which is their difference. This is impossible. Therefore, a new prime can always be found, given any set of primes. This proof is attributed to Euclid (Euclid's Elements Proposition 20, Book IX).**

**Problem 28**: Determine the value of n for which the number 127 is a Mersenne prime .
**Ans: If the number M _{n} = 2^{n} - 1 is a prime, then it is a Mersenne prime. 127 = 2^{7} - 1. So 127 is a Mersenne prime for n = 7.**

**Problem 29**: suppose the binomial number N = a^{3} + b^{3} = 9, where a and b are integers. What is the value of a when b = 1?
**Ans: the algebraic expansion of the binomal a ^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2}) = 9. When b = 1, the expression becomes (a+1)(a^{2} -a + 1) = 9. Therefore, (a+1) is either 1, or 3 or 9. The equation is true only when (a+1) = 3. Therefore, a = 2.**

**Problem 30**: all primes > 2 are odd numbers. Therefore, the minimum distance between any two consecutive primes is 2. Pairs of primes with this minimum distance between them are called *prime twins*. Suppose p_{1} and p_{2} are prime twins and p_{2} and p_{3} are prime twins. What is the sum of the even number between (p_{1}, p_{2}) and the even number between (p_{2}, p_{3}) if p_{2} = 19
**Ans: 38. **

Peter Oye Sagay