Class 10 Maths NCERT Solutions Books

Class 10 maths Integers: Integers are classified into negative integers, zero and positive integers, integers can be classified as prime numbers and composite numbers.

Rational Numbers: A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number.

Real Numbers: The totality of rational numbers and irrational numbers is called the real numbers. i.e. rational numbers and irrational numbers taken together are called Real Numbers.

Prime Number : A natural number is said to be prime, if it has only two different(distinct) factors namely one and itself. We say 4 and 9 are not prime, whereas 2, 3, 5, 7, 11 are prime numbers.

• • • •

1 is not a prime number.

The only even prime number is 2.

The smallest prime number is 2.

All prime numbers are odd except 2.

Twin-Prime: A pair of prime numbers is said to be twin-prime if they differ by 2.

For example {3, 5}, {11, 13}, {17, 19}, {29, 31}, {29, 31}, {41, 43}, {71, 73} are all twin-prime.

Composite Number: A natural number is said to be composite, if it has atleast three different factors. 4, 6, 12, are all composite numbers.

• • •

1 is not a composite number.

4 is the smallest composite number.

1 is neither prime nor composite number.

Co-prime: A pair of numbers is said to be co-prime, if the numbers have no common factor other than one. i.e., Two numbers are said to be co-prime if and only if their highest common factor is 1. i.e., Two numbers are co-prime or relatively prime, if there is no factor common to them except 1.

For example: The pairs of number (2, 3), (5, 9), (7, 13), (12, 17), (12, 35), (63, 26), (162, 35), etc are co-prime and the pairs of numbers (4, 6), (45, 65), (25, 125), (60, 75) are not co-prime.

Perfect Number: A number is said to be perfect, if it is equal to the sum of its factors other than itself.

For example: 6 = (1 + 2 + 3); 28 = (1 + 2 + 4 + 7 + 14) therefore 6 and 28 are perfect numbers.

Even Numbers: A natural number is said to be even if it is multiple of 2 or it is divisible by 2 2, 4, 6, 8, 10, 12 ……. are examples of even numbers.

Odd Numbers: A natural number is said to be odd, if it is not even or if it is not divisible by 2.

EUCLID’S DIVISION LEMMA: Dividend = divisor × quotient + remainder.

Given two positive integers a and b. There exist unique integers q and r satisfying a = bq + r ,

where 0 £ r < b

Where a is dividend, b is divisor, q is quotient and r is remainder.

The most important application to this algorithm is to find HCF of two given positive integers.

If we divide 119 by 8, we get 14 as quotient and 7 as remainder. \

119 = (8 × 14) + 7

EUCLID’S DIVISION ALGORITHM: To obtain the HCF of two positive integers, say c and d, with c>d, follow the steps below: Step 1:

Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and r such that c = dq + r, 0 £r<d.

Step 2:

If r = 0, d is the HCF of c and d. If r ¹ 0, apply the division lemma to d and r.

Step 3:

Write d = er + r1 where 0 < r1 < r

Step 4:

Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

To find HCF of 575 and 15. Let us use Euclid’s algorithm 575 = 15 × 38 + 5 Now, consider 15 and 5 and applying Euclid’s algorithm again.

15 = 5 × 3 + 0 Here, the remainder is zero. \

HCF of 15 and 5 is 5.

\

HCF of 575 and 15 is also 5.