# Higher Algebra Hall & Knight Theory of Numbers (Chapter 30) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 30 ‘Theory of Numbers’ are prepared to help you in solving the advanced level questions in this chapter. These solutions are designed as per the latest syllabus for JEE and NEET and will help you learn the statement of principles. You will be able to revise some important results such as the number of primes is infinite, no rational algebraical formula can represent primes only, and that any number can be disintegrated into prime factors in only one way. You will also learn to evaluate the number of divisors of an integer, ways to resolve integers into two factors, the sum of the divisors and the highest power of prime in n! with the help of our solutions.

Higher Algebra by Hall & Knight has two exercises covering a total of 59 questions. Solving these exercise questions will impart conceptual clarity and efficient time management skills. The exercise questions are crucial if you aim at appearing in competitive exams like JEE or NEET. Our solutions will help you understand the product of r consecutive integers that are divisible by r! and Fermat’s Theorem. Other topics covered solutions for Theory of Numbers in Higher Algebra by H.S. Hall and S.R. Knight include an introduction to congruency, some important results, Wilson’s Theorem and its proofs by induction.

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## Important Topics Covered under Hall and Knight Higher Algebra Solutions Chapter 30: Theory of Numbers

**Introduction**

- The word ‘number’ in all the instances in the discussion of this chapter is used to describe a positive integer.
- Prime numbers are the numbers which are not divisible by any other number other than 1 and the number itself. The numbers that are divisible by numbers other than 1 and itself are termed as composite numbers.

**Important Results**

- Let us consider a number ‘a’ which divides the product of b and c, that is, bc. If ‘a’ is not a factor of b, then ‘a’ will be certainly the factor of c.
- Let us consider a number ‘a’ which divides the product b.c.d, then the prime number ‘a’ must divide one of the numbers. In that order, if a divides bn then a is a factor of b.
- If a is prime to b and c, it will also be prime to the product of the two bc. Also, if a is prime to bc, it will imply that a is prime to b and c separately.
- If a is prime to band vice versa, then every positive integral power of
*a*will remain prime to each positive integral power of b. - If a happens to be prime to b then the fractionsandwill be in their lowest possible terms, where n and m are positive integers.
- There is an unlimited number of prime numbers.
- Only prime numbers can never be represented by a given rational algebraical formula.
- There is only one way that a given number can be resolved into the prime factors.
- The number of ways in which a given composite number, N can be resolved into 2 factors, where N=
*a*^{p }b^{q}c^{r}. If N is not a perfect square then the required number isis a perfect square, then the required number is - Sum of the divisors of a number: Let the number be N=a
^{p}b^{q}c^{r}then the sum is - The product of ‘r’ consecutive integers is divisible by r!

### Exercise Discussion of Hall and Knight Higher Algebra Solutions Chapter 30:Theory of Numbers

- There are 2 exercises in this chapter providing comprehensive coverage of all the topics in this chapter.
- Exercise 30.a consists of questions that are based on the topics, to find the number of divisors of an integer, ways to resolve integers into two factors, the sum of the divisors, highest power of prime in n! and product of r consecutive integers that are divisible by r!
- Exercise 30.b consists of a total of 26 questions. These questions are based on congruency, Wilson’s Theorem, and Fermat Theorem. You will be required to show the divisibility of given algebraic expressions by certain numbers. You will need a thorough knowledge of prime numbers, factors, and multiplication rules to solve this exercise with completely clear concepts in your head.

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