Q1: Given that HCF (150, 100) = 50. Find LCM (150, 100).
Sol: LCM × HCF = Product of the two numbers
∴ 150 × 100 = LCM × HCF
⇒ LCM × 50 = 150 × 100
⇒
Q2: Given that LCM (26, 91) = 182. Find their HCF.
Sol: ∵ HCF × LCM = Product of the two numbers
∴ HCF × 182 = 26 × 91
⇒
Q3: The LCM and HCF of the two numbers are 240 and 12 respectively. If one of the numbers is 60, then find the other number.
Sol: Let the required number be ‘x’.
∵ LCM × HCF = Product of the two numbers
∴ 60 × x = 240 × 12
⇒
Q4: The decimal expansion of the rational number, will terminate after how many places of decimal?
Sol:
Thus, will terminate after 4 places of decimal.
Q5: What is the exponent of 3 in the prime factorisation of 864.
Sol:
Making prime factors of 864. ,⇒ 864 = 3 × 3 × 3 × 2 × 2 × 2 × 2 × 2= 3^{3} × 2^{5}
∴ Exponent of 3 in prime factorisation of 864 = 3.
Q6: State the fundamental theorem of arithmetic.
Sol: Every composite number can be expressed as the product of primes and this decomposition is unique apart from the order in which prime factors occur.
Q7: Define an irrational number.
Sol: Those numbers which neither terminate in their decimal expansion nor can be expressed as recurring decimals are irrational numbers i.e., the numbers which cannot be expressed as p/q form (q ≠ 0), are called irrational numbers.
Q8: Write the condition for a rational number which can have a terminating decimal expansion.
Sol: A rational number x = p/q can have a terminating decimal expansion if the prime factorisation of q is of the form of 2^{n} · 5^{m}, where m and n are nonnegative integers.
Q9: Write the condition for a rational number which has a nonterminating repeating decimal expansion.
Sol: A rational number x = p/q can have a nonterminating repeating decimal expansion if the prime factorisation of q is not of the form 2^{n} · 5^{m}, where n, m are nonnegative integers.
Q10: Can two numbers have 24 as their HCF and 7290 as their LCM? Give reasons.
Sol: No, because HCF always divides LCM but here 24 does not divide 7290.
Q11: If 6^{n} is a number such that n is a natural number. Check whether there is any value of n ∈ N for which 6^{n} is divisible by 7.
Sol: ∵ 6 = 2 × 3
∴ 6^{n} = (2 × 3)^{n} = 2^{n} × 3^{n}
i.e., the prime factorisation of 6^{n} does not contain the prime number 7 thus the number 6^{n} is not divisible by 7.
Q12: Write 98 as the product of its prime factors.
Sol: ∵
∴ 98 = 2 × 7 × 7 ⇒ 98 = 2 × 7^{2}
Q13: Without actually performing the long division, state whether will have a terminating or nonterminating repeating decimal expansion.
Sol: Let =
∵ Prime factors of q are not of the for 2^{n} · 5^{m}.
∴ will have a nonterminating repeating decimal expansion.
Q14: Without actually performing the long division, state whether 17/3125 will have a terminating decimal expansion or a nonterminating repeating decimal expansion.
Sol: ∵ The denominator of 17/3125 is given by
3125 = 5 × 5 × 5 × 5 × 5
= 1 × 55
= 2^{0 }× 5^{5} ∵ 2^{0} = 1
∴
i.e., 17/3125 is a terminating decimal.
Q15: Express 156 as a product of its prime factors.
Sol: ∵ 156 = 2 × 78
= 2 × 2 × 39
= 2 × 2 × 3 × 13
∴ 156 = 2^{2} × 3 × 13
Q16: If the product of two numbers is 20736 and their LCM is 384, find their HCF.
Sol: ∵ LCM × HCF = Product of two numbers
∴ 384 × HCF = 20736
⇒ HCF = 20736 /384 = 54.
Q17: Find the LCM and HCF of 120 and 144 by the Fundamental Theorem of Arithmetic.
Sol: We have 120 = 2 × 2 × 2 × 3 × 5 = 2^{3 }× 3 × 5
144 = 2 × 2 × 2 × 2 × 3 × 3 = 2^{4} × 3^{2}
∴ LCM = 2^{4} × 3^{2} × 5 = 720
HCF = 2^{3} × 3 = 24
Q18: Find the HCF × LCM for the numbers 100 and 190.
Solution: HCF × LCM = 1^{st} Number × 2^{nd }Number
= 100 × 190 = 19000.
Q19: Find the (HCF × LCM) for the numbers 105 and 120.
Solution: HCF × LCM = 1^{st} number × 2^{nd number}
= 105 × 120 = 12600.
Q20: Write a rational number between √2 and √3.
Sol: ∵ √2 = 1.41 ..... and
√3 = 1.73 .....
∴ one rational number between 1.41 .....and 1.73 ..... is 1.5
i.e., one rational number between √2 and √3 is 1.5.
116 videos420 docs77 tests

1. What are real numbers? 
2. How can you determine if a number is a real number? 
3. What is the difference between rational and irrational numbers in the context of real numbers? 
4. How do real numbers relate to everyday life? 
5. Can you give examples of irrational numbers? 

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