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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.
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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.
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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.
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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.
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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/