What do you mean by Euclid’s division algorithm.
Answer: Euclid’s division algorithm: The Euclid’s division algorithm states that for any two positive integers a and b, there exist unique whole numbers q and r, such that a = b × q + r where 0 ≤ r...
State fundamental theorem of arithmetic?
Answer: The fundamental theorem of arithmetic, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and this product is unique.
Express 360 as product of its prime factors
Answer: Using prime factorization: 360 = 23 × 32 × 5
If a and b are two prime numbers then find the HCF (a, b)
Answer: Using prime factorization, a = a b = b HCF = 1 Hence, HCF(a,b) = 1
If a and b are two prime numbers then find the HCF(a, b)
Answer: Using prime factorization, a = a b = b LCM = a × b Hence, LCM (a, b) = ab
The product of two numbers is 1050 and their HCF is 25. Find their LCM.
Answer: HCF of two numbers = 25 Product of two numbers = 1050 Consider, LCM be x. Product of two numbers = HCF × LCM 1050 = 25 × x x = 1050/25 x = 42 LCM is 42.
What is a composite number?
Answer: A composite number is a positive integer which is not prime such as that has factors other than 1 and itself.
If a and b are relatively prime then what is their HCF?
Answer: If two numbers are relatively prime then their greatest common factor will be 1. HCF (a, b) = 1.
If the rational number a/b has a terminating decimal expansion, what is the condition to be satisfied by b?
Answer: Consider, x - rational number whose decimal expansion terminates. Express x in the form ????/????, where a and b are coprime Prime factorization of b is of the form (2m × 5n) [m and n -...
Find the simplest form of
Answer: $\begin{array}{l} \frac{{2\sqrt {45} + 3\sqrt {20} }}{{2\sqrt 5 }} = \frac{{2\sqrt {3 \times 3 \times 5} + 3\sqrt {2 \times 2 \times 5} }}{{2\sqrt 5 }}\\ = > \frac{{2 \times \sqrt 5 +...
Write the decimal expansion of
Answer: $\begin{array}{l} \frac{{73}}{{({2^4} \times {5^3})}} = \frac{{73 \times 5}}{{{2^4} \times {5^4}}}\\ = > \frac{{365}}{{{{(2 \times 5)}^4}}}\\ = > \frac{{365}}{{{{(10)}^4}}}\\ = >...
Show that there is no value of n for which HCF if 25 and LCM is 520?
Answer: Consider, (2n × 5n) = (2 × 5)n => 10n For any value of n, we get 0 in the end. Hence, there is no value of n for which (2n × 5n) ends in 5.
Is it possible to have two numbers whose HCF if 25 and LCM is 520?
Answer: No, it is not possible to have two numbers whose HCF is 25 and LCM is 520. HCF must be a factor of LCM, but 25 is not a factor of 520.
Give an example of two irrationals whose sum is rational.
Answer: Consider, Two irrationals be 4 - √5 and 4 + √5 (4 - √5) + (4 + √5) = 8 Hence, 8 is a rational number.
Give an example of two irrationals whose product is rational.
Answer: Consider, Two irrationals be 4 √5 and 3 √5 (4 √5) × (3 √5) = 60 Hence, 60 is a rational number.
If a and b are relatively prime, what is their LCM?
Answer: If two numbers are relatively prime then their greatest common factor will be 1. ∴ HCF (a,b) = 1 Product of two numbers = HCF × LCM a × b = 1 × LCM ∴ LCM = ab LCM (a,b) is...
The LCM of two numbers is 1200, show that the HCF of these numbers cannot be 500. Why?
Answer: If the LCM of two numbers is 1200 then, it is not possible to have their HCF equals to 500. HCF must be a factor of LCM, but 500 is not a factor of 1200.
Express as a rational number simplest form.
Answer: Consider, x be $0.\overline{4}$ x = $0.\overline{4}$ Multiplying, 10x = $4.\overline{4}$ Subtracting, 10x – x = $4.\overline{4}$ - $0.\overline{4}$ 9x = 4 x = $\frac{4}{9}$ The simplest...
Express as a rational number in simplest form.
Answer: Consider, x be $0.\overline{23}$ x = $0.\overline{23}$ Multiplying, 100x = $23.\overline{23}$ Subtracting, 100x – x = $23.\overline{23}$ - $0.\overline{23}$ 99x = 23 x = $\frac{23}{99}$ The...
Explain why 0.15015001500015……. is an irrational form.
Answer: Irrational numbers are non-terminating non-recurring decimals. Hence, 0.15015001500015…. is an irrational number.
Show that is irrational.
Answer: Consider, $\frac{\sqrt{2}}{3}$ is a rational number, where p and q are some integers and HCF(p,q) = 1. √2q = 3p (√2q)2 = (3p)2 2q2 = 9p2 p2 is divisible by 2 p is divisible by 2 Consider, p...
Write a rational number between √3 and 2
Answer: √3 = 1.732…. Take 1.8 as the required rational number between √3 and 2. The rational number = 1.8
Explain why is a rational number?
Answer: $3.\overline{1416}$ is a non-terminating repeating decimal. Thus, it is a rational number.