From this exercise, we can infer that the diagonal of a rectangle always divides it into two equal triangles. Both triangles are congruent in shape and size, and their areas are equal to half the area of the rectangle. This principle applies universally to all rectangles and squares, regardless of tRead more
From this exercise, we can infer that the diagonal of a rectangle always divides it into two equal triangles. Both triangles are congruent in shape and size, and their areas are equal to half the area of the rectangle. This principle applies universally to all rectangles and squares, regardless of their dimensions, establishing a consistent relationship between a rectangle and its diagonally divided parts.
Twin primes are pairs of prime numbers that differ by 2. Between 1 and 100, the twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73). Each pair consists of two primes satisfying the condition of a difference of 2. These pairs highlight the fascinating patterRead more
Twin primes are pairs of prime numbers that differ by 2. Between 1 and 100, the twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73). Each pair consists of two primes satisfying the condition of a difference of 2. These pairs highlight the fascinating patterns of prime distribution within a specific range.
The statement is true because numbers ending in 4 are even. A prime number has only two divisors: 1 and itself. All even numbers are divisible by 2, making them composite, except for the number 2 itself. Since 2 is the only even prime number, any number with a units digit of 4 cannot be prime. For eRead more
The statement is true because numbers ending in 4 are even. A prime number has only two divisors: 1 and itself. All even numbers are divisible by 2, making them composite, except for the number 2 itself. Since 2 is the only even prime number, any number with a units digit of 4 cannot be prime. For example, numbers like 14, 24, and 34 all have divisors other than 1 and themselves.
The statement is false. A prime number has only two divisors: 1 and itself. The product of two or more primes, such as 2 × 3 = 6, is composite because it has additional divisors beyond 1 and itself (2, 3, and 6). While primes are building blocks for composite numbers, their multiplication always resRead more
The statement is false. A prime number has only two divisors: 1 and itself. The product of two or more primes, such as 2 × 3 = 6, is composite because it has additional divisors beyond 1 and itself (2, 3, and 6). While primes are building blocks for composite numbers, their multiplication always results in numbers with more than two divisors, thereby disqualifying them from being prime.
This statement is false because prime numbers have exactly two factors, which are 1 and the number itself. For example, 13 is a prime number because its only factors are 1 and 13. If a number has no factors other than 1 and itself, it qualifies as prime. The absence of additional factors differentiaRead more
This statement is false because prime numbers have exactly two factors, which are 1 and the number itself. For example, 13 is a prime number because its only factors are 1 and 13. If a number has no factors other than 1 and itself, it qualifies as prime. The absence of additional factors differentiates primes from composite numbers, which have more than two divisors.
The statement is false because not all even numbers are composite. The number 2, the smallest and only even prime number, has exactly two factors: 1 and itself. While other even numbers like 4, 6, and 8 are composite due to having more than two divisors, 2’s unique properties as a prime number makeRead more
The statement is false because not all even numbers are composite. The number 2, the smallest and only even prime number, has exactly two factors: 1 and itself. While other even numbers like 4, 6, and 8 are composite due to having more than two divisors, 2’s unique properties as a prime number make it an exception. Thus, the generalization does not hold for all even numbers.
The statement is true. 2 and 3 are the only consecutive prime numbers because any number immediately after a prime is either even or divisible by smaller primes, thus making it composite. For example, after 5 comes 6 (even), and after 7 comes 8 (even). This pattern ensures that primes apart from 2 aRead more
The statement is true. 2 and 3 are the only consecutive prime numbers because any number immediately after a prime is either even or divisible by smaller primes, thus making it composite. For example, after 5 comes 6 (even), and after 7 comes 8 (even). This pattern ensures that primes apart from 2 and 3 are always separated by at least one composite number, confirming the validity of this observation.
Among the given numbers, 105 and 330 are products of exactly three distinct prime numbers. Their factorizations are: • 105 = 3 × 5 × 7. • 330 = 2 × 3 × 5 × 11, but considering three distinct primes (2, 3, 5), this is valid. Other numbers, like 45 (3² × 5) and 91 (7 × 13), do not meet the condition.Read more
Among the given numbers, 105 and 330 are products of exactly three distinct prime numbers. Their factorizations are:
• 105 = 3 × 5 × 7.
• 330 = 2 × 3 × 5 × 11, but considering three distinct primes (2, 3, 5), this is valid.
Other numbers, like 45 (3² × 5) and 91 (7 × 13), do not meet the condition. The count excludes repetitions and confirms distinct prime contributions.
Using 2, 4, and 5, the six three-digit permutations are: 245, 254, 425, 452, 524, and 542. Checking their primality: • 245 is divisible by 5. • 254 is divisible by 2. • 425 is divisible by 5. • 452 is divisible by 2. • 524 is divisible by 2. • 542 is divisible by 2. Thus, none of these numbers are pRead more
Using 2, 4, and 5, the six three-digit permutations are: 245, 254, 425, 452, 524, and 542. Checking their primality:
• 245 is divisible by 5.
• 254 is divisible by 2.
• 425 is divisible by 5.
• 452 is divisible by 2.
• 524 is divisible by 2.
• 542 is divisible by 2.
Thus, none of these numbers are prime. Despite using each digit exactly once, all numbers are divisible by either 2 or 5.
Several primes satisfy the condition where doubling and adding 1 yields another prime: • For 3, 2 × 3 + 1 = 7 (prime). • For 5, 2 × 5 + 1 = 11 (prime). • For 11, 2 × 11 + 1 = 23 (prime). • For 13, 2 × 13 + 1 = 27 (prime). The sequence demonstrates how doubling primes can yield new primes, though excRead more
Several primes satisfy the condition where doubling and adding 1 yields another prime:
• For 3, 2 × 3 + 1 = 7 (prime).
• For 5, 2 × 5 + 1 = 11 (prime).
• For 11, 2 × 11 + 1 = 23 (prime).
• For 13, 2 × 13 + 1 = 27 (prime).
The sequence demonstrates how doubling primes can yield new primes, though exceptions exist. Testing higher primes verifies these conditions.
Can you draw any inferences from this exercise?
From this exercise, we can infer that the diagonal of a rectangle always divides it into two equal triangles. Both triangles are congruent in shape and size, and their areas are equal to half the area of the rectangle. This principle applies universally to all rectangles and squares, regardless of tRead more
From this exercise, we can infer that the diagonal of a rectangle always divides it into two equal triangles. Both triangles are congruent in shape and size, and their areas are equal to half the area of the rectangle. This principle applies universally to all rectangles and squares, regardless of their dimensions, establishing a consistent relationship between a rectangle and its diagonally divided parts.
For more NCERT Solutions for Class 6 Math Chapter 6 Perimeter and Area Extra Questions and Answer:
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Find twin primes between 1 and 100.
Twin primes are pairs of prime numbers that differ by 2. Between 1 and 100, the twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73). Each pair consists of two primes satisfying the condition of a difference of 2. These pairs highlight the fascinating patterRead more
Twin primes are pairs of prime numbers that differ by 2. Between 1 and 100, the twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73). Each pair consists of two primes satisfying the condition of a difference of 2. These pairs highlight the fascinating patterns of prime distribution within a specific range.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Identify whether each statement is true or false. Explain. a) There is no prime number whose units digit is 4.
The statement is true because numbers ending in 4 are even. A prime number has only two divisors: 1 and itself. All even numbers are divisible by 2, making them composite, except for the number 2 itself. Since 2 is the only even prime number, any number with a units digit of 4 cannot be prime. For eRead more
The statement is true because numbers ending in 4 are even. A prime number has only two divisors: 1 and itself. All even numbers are divisible by 2, making them composite, except for the number 2 itself. Since 2 is the only even prime number, any number with a units digit of 4 cannot be prime. For example, numbers like 14, 24, and 34 all have divisors other than 1 and themselves.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Identify whether each statement is true or false. Explain. b) A product of primes can also be prime.
The statement is false. A prime number has only two divisors: 1 and itself. The product of two or more primes, such as 2 × 3 = 6, is composite because it has additional divisors beyond 1 and itself (2, 3, and 6). While primes are building blocks for composite numbers, their multiplication always resRead more
The statement is false. A prime number has only two divisors: 1 and itself. The product of two or more primes, such as 2 × 3 = 6, is composite because it has additional divisors beyond 1 and itself (2, 3, and 6). While primes are building blocks for composite numbers, their multiplication always results in numbers with more than two divisors, thereby disqualifying them from being prime.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Identify whether each statement is true or false. Explain. c) Prime numbers do not have any factors.
This statement is false because prime numbers have exactly two factors, which are 1 and the number itself. For example, 13 is a prime number because its only factors are 1 and 13. If a number has no factors other than 1 and itself, it qualifies as prime. The absence of additional factors differentiaRead more
This statement is false because prime numbers have exactly two factors, which are 1 and the number itself. For example, 13 is a prime number because its only factors are 1 and 13. If a number has no factors other than 1 and itself, it qualifies as prime. The absence of additional factors differentiates primes from composite numbers, which have more than two divisors.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Identify whether each statement is true or false. Explain. d) All even numbers are composite numbers.
The statement is false because not all even numbers are composite. The number 2, the smallest and only even prime number, has exactly two factors: 1 and itself. While other even numbers like 4, 6, and 8 are composite due to having more than two divisors, 2’s unique properties as a prime number makeRead more
The statement is false because not all even numbers are composite. The number 2, the smallest and only even prime number, has exactly two factors: 1 and itself. While other even numbers like 4, 6, and 8 are composite due to having more than two divisors, 2’s unique properties as a prime number make it an exception. Thus, the generalization does not hold for all even numbers.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Identify whether each statement is true or false. Explain. e) 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.
The statement is true. 2 and 3 are the only consecutive prime numbers because any number immediately after a prime is either even or divisible by smaller primes, thus making it composite. For example, after 5 comes 6 (even), and after 7 comes 8 (even). This pattern ensures that primes apart from 2 aRead more
The statement is true. 2 and 3 are the only consecutive prime numbers because any number immediately after a prime is either even or divisible by smaller primes, thus making it composite. For example, after 5 comes 6 (even), and after 7 comes 8 (even). This pattern ensures that primes apart from 2 and 3 are always separated by at least one composite number, confirming the validity of this observation.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
Among the given numbers, 105 and 330 are products of exactly three distinct prime numbers. Their factorizations are: • 105 = 3 × 5 × 7. • 330 = 2 × 3 × 5 × 11, but considering three distinct primes (2, 3, 5), this is valid. Other numbers, like 45 (3² × 5) and 91 (7 × 13), do not meet the condition.Read more
Among the given numbers, 105 and 330 are products of exactly three distinct prime numbers. Their factorizations are:
• 105 = 3 × 5 × 7.
• 330 = 2 × 3 × 5 × 11, but considering three distinct primes (2, 3, 5), this is valid.
Other numbers, like 45 (3² × 5) and 91 (7 × 13), do not meet the condition. The count excludes repetitions and confirms distinct prime contributions.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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How many three-digit prime numbers can you make using each of 2, 4, and 5 exactly once?
Using 2, 4, and 5, the six three-digit permutations are: 245, 254, 425, 452, 524, and 542. Checking their primality: • 245 is divisible by 5. • 254 is divisible by 2. • 425 is divisible by 5. • 452 is divisible by 2. • 524 is divisible by 2. • 542 is divisible by 2. Thus, none of these numbers are pRead more
Using 2, 4, and 5, the six three-digit permutations are: 245, 254, 425, 452, 524, and 542. Checking their primality:
• 245 is divisible by 5.
• 254 is divisible by 2.
• 425 is divisible by 5.
• 452 is divisible by 2.
• 524 is divisible by 2.
• 542 is divisible by 2.
Thus, none of these numbers are prime. Despite using each digit exactly once, all numbers are divisible by either 2 or 5.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime?
Several primes satisfy the condition where doubling and adding 1 yields another prime: • For 3, 2 × 3 + 1 = 7 (prime). • For 5, 2 × 5 + 1 = 11 (prime). • For 11, 2 × 11 + 1 = 23 (prime). • For 13, 2 × 13 + 1 = 27 (prime). The sequence demonstrates how doubling primes can yield new primes, though excRead more
Several primes satisfy the condition where doubling and adding 1 yields another prime:
• For 3, 2 × 3 + 1 = 7 (prime).
• For 5, 2 × 5 + 1 = 11 (prime).
• For 11, 2 × 11 + 1 = 23 (prime).
• For 13, 2 × 13 + 1 = 27 (prime).
The sequence demonstrates how doubling primes can yield new primes, though exceptions exist. Testing higher primes verifies these conditions.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
See lesshttps://www.tiwariacademy.com/ncert-solutions-class-6-maths-ganita-prakash-chapter-5/