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= 33 × 25
∴ 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 2n · 5m, where m and n are non-negative integers.
Q9: Write the condition for a rational number which has a non-terminating repeating decimal expansion.
Sol: A rational number x = p/q can have a non-terminating repeating decimal expansion if the prime factorisation of q is not of the form 2n · 5m, where n, m are non-negative 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 6n is a number such that n is a natural number. Check whether there is any value of n ∈ N for which 6n is divisible by 7.
Sol: ∵ 6 = 2 × 3
∴ 6n = (2 × 3)n = 2n × 3n
i.e., the prime factorisation of 6n does not contain the prime number 7 thus the number 6n is not divisible by 7.
Q12: Write 98 as the product of its prime factors.
Sol: ∵
∴ 98 = 2 × 7 × 7 ⇒ 98 = 2 × 72
Q13: Without actually performing the long division, state whether will have a terminating or non-terminating repeating decimal expansion.
Sol: Let =
∵ Prime factors of q are not of the for 2n · 5m.
∴ will have a non-terminating repeating decimal expansion.
Q14: Without actually performing the long division, state whether 17/3125 will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Sol: ∵ The denominator of 17/3125 is given by
3125 = 5 × 5 × 5 × 5 × 5
= 1 × 55
= 20 × 55 |∵ 20 = 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 = 22 × 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 = 23 × 3 × 5
144 = 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32
∴ LCM = 24 × 32 × 5 = 720
HCF = 23 × 3 = 24
Q18: Find the HCF × LCM for the numbers 100 and 190.
Solution: HCF × LCM = 1st Number × 2nd Number
= 100 × 190 = 19000.
Q19: Find the (HCF × LCM) for the numbers 105 and 120.
Solution: HCF × LCM = 1st number × 2nd 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.
124 videos|457 docs|77 tests
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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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