GAJERA INTERNATIONAL SCHOOL, SURAT.

SUBJECT: MATHS (Study Material)

Date: 24/03/2020 Std: - X Topic : Real Numbers

Concepts

1. Euclid's Division Lemma 2. Euclid's Division Algorithm

3. Prime Factorization

4. Fundamental Theorem of Arithmetic

5. Decimal expansion of rational numbers

A dividend can be written as, Dividend = Divisor × Quotient + Remainder. This brings to

Euclid's division lemma.

Euclid’s division lemma: Euclid’s division lemma, states that for any two positive integers ‘a’ and ‘b’ we can find two whole numbers ‘q’ and ‘r’ such that a = b × q + r where 0 ≤ r < b.

Euclid’s division lemma can be used to find the highest common factor of any two positive

integers and to show the common properties of numbers.

The following steps to obtain H.C.F using Euclid’s division lemma:

1. Consider two positive integers ‘a’ and ‘b’ such that a > b. Apply Euclid’s division lemma to the

given integers ‘a’ and ‘b’ to find two whole numbers ‘q’ and ‘r’ such that, a = b x q + r.

2. Check the value of ‘r’. If r = 0 then ‘b’ is the HCF of the given numbers. If r ≠ 0, apply

Euclid’s division lemma to find the new divisor ‘b’ and remainder ‘r’.

3. Continue this process till the remainder becomes zero. In that case the value of the

divisor ‘b’ is the HCF (a , b). Also HCF(a ,b) = HCF(b, r).

Example 3:

Example 4:

Fundamental Theorem of Arithmetic: Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed or factorized as a unique product of prime numbers except in the order of the prime factors.

We can write the prime factorization of a number in the form of powers of its prime factors.

By expressing any two numbers as their prime factors, their highest common factor (HCF) and

lowest common multiple (LCM) can be easily calculated.

The HCF of two numbers is equal to the product of the terms containing the least powers of

common prime factors of the two numbers.

The LCM of two numbers is equal to the product of the terms containing the greatest powers of

all prime factors of the two numbers.

For any two positive integers a and b, HCF(a , b) x LCM(a , b) = a x b.

For any three positive integers a, b and c,

LCM(a, b, c) = a.b.c HCF(a, b, c) HCF(a, b).HCF(b, c).HCF(c, a)

HCF(a, b, c) = a.b.c LCM(a, b, c) LCM(a, b).LCM(b, c).LCM(c, a) .

Rational number:

A number which can be written in the form ⁄where a and b are integers and b ≠ 0 is called a

rational number. Rational numbers are of two types depending on whether their decimal form is terminating or

recurring.

Irrational number:

A number which cannot be written in the form ⁄, where a andb are integers and b ≠ 0 is called

an irrational number. Irrational numbers which have non-terminating and non-repeating decimal representation.

The sum or difference of a two irrational numbers is also rational or an irrational number.

The sum or difference of a rational and an irrational number is also an irrational number.

Product of a rational and an irrational number is also an irrational number.

Product of a two irrational numbers is also rational or an irrational number.

Example 2:

Example:

Example:

Example:

Decimal expansion of rational numbers:

Theorem:

Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

Theorem:If p q is a rational number, such that the prime factorisation of q is of the form 2a5b,

where a and b are positive integers, then the decimal expansion of the rational number p q

terminates.

Theorem: If a rational number is a terminating decimal, it can be written in the form p q , where p and q are co prime and the prime factorisation of q is of the form 2a5b, where a and b are positive integers.

Theorem: If p q is a rational number such that the prime factorisation of q is not of the form 2a5b

where a and b are positive integers, then the decimal expansion of the rational number p q does

not terminate and is recurring.

Note: The product of the given numbers is equal to the product of their HCF and LCM. This

result is true for all positive integers and is often used to find the HCF of two given numbers if

their LCM is given and vice versa.

Example:

Example:

Example:

Maths Work Sheet Class - X Chapter: - Real Numbers

Q01 :} Find the smallest number which when divided by 30, 40 and 60 leaves the

remainder 7 in each case.

Q02 :} The dimensions of a room are 6 m 75 cm, 4 m 50 cm and 2 m 25 cm. Find the

length of the largest measuring rod which can measure the dimensions in exact number

of times.

Q03 :} The HCF of 2 numbers is 75 and their LCM is 1500. If one of the numbers is

300, find the other.

Q04 :} Prove that is irrational.

Q05 :} Can 72 and 20 respectively be the LCM and HCF of two numbers. Write

down the reason.

Q06 :} If a and b are two prime numbers, write their HCF and LCM.

Q07 :} If p and q are two coprime numbers, write their HCF and LCM.

Q08 :} Without actual division, state whether the decimal form of is

terminating OR recurring.

Q09 :} Find the HCF and LCM of 350 and 400 and verify that

HCFxLCM=Product of the numbers.

Q10 :} Simplify:

Q11 :} Write down 5 irrational numbers in radical form which are lying between

4 and 5.

Q12 :} Write down 2 rational numbers lying between and .

Q13 :} Complete the missing entries in the following factor tree.

2

11

5

Q14 :} Prove that is irrational if p and q are prime numbers.

Q15 :} Find the largest number which divides 245 and 1205 leaving the

remainder

5 in each case.

Q16 :} Find the largest number which divides 303, 455 and 757

leaving the remainder 3, 5 and 7 respectively.

Q17 :} Prove that is irrational.

Q18 :} Prove that is irrational.

Q19 :} Find the HCF and the LCM of the following by prime

factorization. a) 360 , 756

b) 2x4y

3z , 32x

3y

4p

2

Q20 :} Find the HCF by Euclid's Division

Algorithm. a) 256 , 352

b) 450 , 500 , 625

Q21 :} Explain why 7x11x13+13 is a composite number.

Q22 :} Show that any positive odd number is of the form 6q + 1, 6q + 3

or 6q + 5, where q is an integer.

Q23 :} Show that the square of any positive integer is of the form 3m or

3m + 1, where m is an integer.

Q24 :} Use Euclid's division lemma to show that the cube of any positive

integer is of the form 9m, 9m + 1, 9m + 8, where m is an integer.

Q25 :} There are 3 consecutive traffic lights which turn "green" after

every 36, 42 and 72 seconds. They all were at "green" at 9:00 AM. At

what time will they all turn "green" simultaneously?

Prepared By:-

Deep Shah (Maths Educator – Classes IX & X)