Using Euclid's division algorithm, find the largest number that divides 1251 , 9377 and 15628 leaving remainders 1 , 2 and 3 , respectively.

Since, 1 , 2 and 3 are the remainders of 1251 , 9377 and 15628 , respectively. Thus, after subtracting these remainders from the numbers. We have the numbers, 1251 - 1 = 1250 , 9377 - 2 = 9375 and 15628 - 3 = 15625 which is divisible by the required number. Now, required number = H C F of 1250 , 9375 and 15625 [for the largest number] By Euclid's division algorithm, a = b q + r . . . i [ ∵ dividend = divisor × quotient + remainder] For largest number, put a = 15625 and b = 9375 15625 = 9375 × 1 + 6250 [from Eq. i ] ⇒ 9375 = 6250 × 1 + 3125 ⇒ 6250 = 3125 × 2 + 0 So, the H C F of 15625 and 9375 is 3125 . Now, we will find the H C F of 3125 and 1250 . ⇒ 3125 = 1250 × 2 + 625 ⇒ 1250 = 625 × 2 + 0 ∴ H C F 1250 , 9375 , 15625 = 625 Hence, 625 is the largest number which divides 1251 , 9377 and 15628 leaving remainder 1 , 2 and 3 , respectively.

Prove that 3 + 5 is irrational.

Let us suppose that 3 + 5 is rational. Let 3 + 5 = a , where a is rational. Therefore, 3 = a - 5 On squaring both sides, we get ( 3 ) 2 = ( a − 5 ) 2 ⇒ 3 = a 2 + 5 − 2 a 5 [ ∵ ( a − b ) 2 = a 2 + b 2 − 2 a b ] ⇒ 2 a 5 = a 2 + 2 Therefore, 5 = a 2 + 2 2 a which is contradiction. As the right hand side is rational number while 5 is irrational. Since, 3 and 5 are prime number. Hence, 3 + 5 is irrational.

Write the denominator of rational 257 5000 in the form 2 m × 5 n , where m , n are non-negative integers. Hence, write its decimal expansion, without actual division.

Denominator of the rational number 257 5000 is 5000 . Now, factors of 5000 = 2 × 2 × 2 × 5 × 5 × 5 × 5 = ( 2 ) 3 × ( 5 ) 4 , which is of the type 2 m × 5 n , where m = 3 and n = 4 are non-negative integers. ∴ Rational number = 257 5000 = 257 2 3 × 5 4 × 2 2 [since, multiplying numerator and denominator by 2 ] 514 2 4 × 5 4 = 514 10 4 514 10000 = 0 . 0514 Hence, which is the required decimal expansion of the rational 257 5000 and it is also a terminating decimal number.

Prove that p + q is irrational, where p and q are primes.

Let us suppose that p + q is rational. Again, let p + q = a , where a is rational. Therefore, q = a − p On squaring both sides, we get q = a 2 + p − 2 a p [ ∵ ( a − b ) 2 = a 2 + b 2 − 2 a b ] Therefore, p = a 2 + p - q 2 a , which is a contradiction as the right hand side is rational number while p is irrational, since p and q are prime number. Hence, p + q is irrational.

E M B I B E

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise

Neha Tyagi and Amit Rastogi Solutions for Chapter: Real Numbers, Exercise 3: Exercise

Author:Neha Tyagi & Amit Rastogi

Neha Tyagi Mathematics Solutions for Exercise - Neha Tyagi and Amit Rastogi Solutions for Chapter: Real Numbers, Exercise 3: Exercise

Attempt the practice questions on Chapter 1: Real Numbers, Exercise 3: Exercise with hints and solutions to strengthen your understanding. NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS solutions are prepared by Experienced Embibe Experts.

Questions from Neha Tyagi and Amit Rastogi Solutions for Chapter: Real Numbers, Exercise 3: Exercise with Hints & Solutions

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 7

Prove that, if $x$ and $y$ are both odd positive integers, then $x^{2} + y^{2}$ is even but not divisible by $4$ .

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 8

Use Euclid's division algorithm to find out the $H C F$ of $441, 567$ and $693 .$

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 9

Using Euclid's division algorithm, find the largest number that divides $1251, 9377$ and $4$ leaving remainders $1, 2$ and $3$ , respectively.

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 10

Prove that $\sqrt{3} + \sqrt{5}$ is irrational.

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 11

Show that $12^{n}$ cannot end with the digit $0$ or $5$ for any natural number $n$ .

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 12

On a morning walk, three persons step off together and their steps measure $40 cm, 42 cm$ and $45 cm$ , respectively. What is the minimum distance each should walk, so that each can cover the same distance in complete steps?

MEDIUM

10th CBSE

IMPORTANT

NCERT EXEMPLAR PROBLEMS-SOLUTIONS MATHEMATICS>Chapter 1 - Real Numbers>Exercise>Q 13

Write the denominator of rational $\frac{257}{5000}$ in the form $2^{m} \times 5^{n}$ , where $m, n$ are non-negative integers. Hence, write its decimal expansion, without actual division.