# UP Board Solutions for Class 10 Maths Chapter 1 Real Numbers

## Real Numbers

### 1. Euclid’s Algorithm for HCF:

Euclid, a Greek mathematician, devised an interesting method to find the HCF of two numbers. This method is known as Euclid’s Algorithm. An algorithm is a stepwise process to solve a problem.

#### Steps taken to find the HCF of two numbers are as under:

1. Identify the greater number of the two.
2. Consider the greater number as dividend and the smaller one as divisor.
3. Find the quotient and the remainder.
4. If the remainder is zero, then the divisor is the required HCF, otherwise go to the next step.
5. Take the remainder as the new divisor and the divisor as the new dividend.
6. Repeat steps 3, 4 and 5 till the remainder zero is obtained. The last divisor, for which the remainder is zero, is the required HCP.

### 2. Prime Number:

A number is called a prime number if it has no factor other than 1 and the number itself.

### 3. Composite Number:

A number is called a composite number if it has at least one factor other than 1 and the number itself.

Note: 1. It should be noted that the number 1 is neither prime nor composite.
2. 2 is the smallest prime number.
3. 2 is the only even prime number. All other even numbers are composite numbers.
4. If a number is not divisible by any one of the primes less than half of it, then it is prime, otherwise it is composite.

### 4. Fundamental Theorem of Arithmetic:

Every composite number can be expressed as a product of primes, and this decomposition is unique, apart from the order in which the prime factors occur.
Thus, it follows that whether a number is prime or composite, it can be uniquely expressed as the product of prime factors.

### 5. Highest Common Factor:

The HCF of two or more numbers is the greatest among common factors.
In other words, the HCF of two or more numbers is the largest number that divides all the given numbers exactly. It is also called the greatest common divisor (GCD).

#### Prime Factorisation Method to find HCF

1. Find prime factorisation of each of the given numbers.
2. Identify common prime factors.
3. Find the product of all the common prime factors, using each common prime factor the least number of times it appears in the prime factorisation of any of the given numbers. The product so obtained is the required HCF.

### 6. Lowest Common Multiple:

The LCM of two or more numbers is the smallest number which is a multiple of each of the numbers.

In other words, the LCM of two or more numbers is the smallest number which is divisible by all the given numbers, i.e., there cannot be a number divisible by the given numbers and smaller than the LCM.

#### Prime Factorisation Method to find LCM

1. Write the prime factorisation of each of the given numbers.
2. Find the product of all different prime factors of the numbers using each common prime factor the greatest number of times it appears in the prime factorisation of any of the numbers. The product so obtained is the required ICM of the given numbers.

### 7. Properties of HCF and LCM of Given Numbers

1. The HCF of given numbers is not greater than any of the numbers.
2. The LCM of given numbers is not less than any of the given numbers.
3. The HCF of given numbers is always a factor of their LCM.
4. The HCF of two co-prime numbers is 1.
5. The LCM of two or more co-prime numbers is equal to their product.
6. If a and b are two numbers, then a × b = HCF (a,b) x LCM (a, b).
8. Every rational number can be expressed as a decimal.
9. The decimal representation of a rational number is either terminating or non-terminating repeating.
10. If the denominator of a rational number, written in standard form, contains no prime factors other than 2 or 5 (or both), then it can be represented as a terminating decimal. Otherwise, it cannot be
represented as a terminating decimal.