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.
To determine co-primality, find the greatest common divisor (GCD): • (30, 45): Not co-prime, GCD = 15. • (57, 85): Co-prime, GCD = 1, as they share no factors other than 1. • (121, 1331): Not co-prime, GCD = 11. • (343, 216): Co-prime, GCD = 1, as no common factors exist. Pairs are co-prime only ifRead more
To determine co-primality, find the greatest common divisor (GCD):
• (30, 45): Not co-prime, GCD = 15.
• (57, 85): Co-prime, GCD = 1, as they share no factors other than 1.
• (121, 1331): Not co-prime, GCD = 11.
• (343, 216): Co-prime, GCD = 1, as no common factors exist.
Pairs are co-prime only if their GCD is 1, ensuring no shared prime factors.
The prime factorization of 225 is 3² x 5² and that of 27 is 3³. For divisibility, the prime factorization of the divisor must be included in the dividend’s factorization. Here, 27 has 3³, while 225 only has 3², making divisibility impossible. Thus, 225 is not divisible by 27 because the power of 3 iRead more
The prime factorization of 225 is 3² x 5² and that of 27 is 3³. For divisibility, the prime factorization of the divisor must be included in the dividend’s factorization. Here, 27 has 3³, while 225 only has 3², making divisibility impossible. Thus, 225 is not divisible by 27 because the power of 3 in 225’s factorization is insufficient to accommodate the full factorization of 27.
The prime factorization of 96 is 2⁵ × 3, and that of 24 is 2³ ×3. Since all factors of 24 are present in 96’s factorization with equal or greater powers, 96 is divisible by 24. Dividing 96 by 24 confirms this: 96÷24 = 4 with no remainder. This demonstrates that the prime factors of 24 are fully inclRead more
The prime factorization of 96 is 2⁵ × 3, and that of 24 is 2³ ×3. Since all factors of 24 are present in 96’s factorization with equal or greater powers, 96 is divisible by 24. Dividing 96 by 24 confirms this: 96÷24 = 4 with no remainder. This demonstrates that the prime factors of 24 are fully included in those of 96, validating divisibility.
The prime factorization of 343 is 7³, indicating it is composed entirely of the prime number 7. Meanwhile, 17 is a prime number with no shared factors with 7. For divisibility, all prime factors of 17 must appear in 343’s factorization, which they do not. Hence, 343 is not divisible by 17, as theirRead more
The prime factorization of 343 is 7³, indicating it is composed entirely of the prime number 7. Meanwhile, 17 is a prime number with no shared factors with 7. For divisibility, all prime factors of 17 must appear in 343’s factorization, which they do not. Hence, 343 is not divisible by 17, as their factorization reveals no overlap or commonality in prime factors.
The prime factorization of 999 is 3³ x 37, and that of 99 is 3² × 11. For divisibility, the prime factors of 99 must appear in 999’s factorization. Here, 3² is present in 3³ and 11 divides evenly into 999. Performing the division confirms this: 999÷99=10. Thus, all factors of 99 are found in 999’s fRead more
The prime factorization of 999 is 3³ x 37, and that of 99 is 3² × 11. For divisibility, the prime factors of 99 must appear in 999’s factorization. Here, 3² is present in 3³ and 11 divides evenly into 999. Performing the division confirms this: 999÷99=10. Thus, all factors of 99 are found in 999’s factorization, verifying divisibility.
To check co-primality, find the prime factorization of both numbers. For 56, it is 2³ × 7 for 63, it is 3² ×7. The shared factor 7 makes their greatest common divisor (GCD) 7, indicating they are not co-prime. Co-prime numbers must have no common factors other than 1, which is not the case here dueRead more
To check co-primality, find the prime factorization of both numbers. For 56, it is 2³ × 7 for 63, it is 3² ×7. The shared factor 7 makes their greatest common divisor (GCD) 7, indicating they are not co-prime. Co-prime numbers must have no common factors other than 1, which is not the case here due to their overlap in the factor 7.
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.
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Which of the following pairs are co-prime: (30, 45), (57, 85), (121, 1331), (343, 216)?
To determine co-primality, find the greatest common divisor (GCD): • (30, 45): Not co-prime, GCD = 15. • (57, 85): Co-prime, GCD = 1, as they share no factors other than 1. • (121, 1331): Not co-prime, GCD = 11. • (343, 216): Co-prime, GCD = 1, as no common factors exist. Pairs are co-prime only ifRead more
To determine co-primality, find the greatest common divisor (GCD):
• (30, 45): Not co-prime, GCD = 15.
• (57, 85): Co-prime, GCD = 1, as they share no factors other than 1.
• (121, 1331): Not co-prime, GCD = 11.
• (343, 216): Co-prime, GCD = 1, as no common factors exist.
Pairs are co-prime only if their GCD is 1, ensuring no shared prime factors.
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Is the first number divisible by the second? Use prime factorization. a) 225 and 27
The prime factorization of 225 is 3² x 5² and that of 27 is 3³. For divisibility, the prime factorization of the divisor must be included in the dividend’s factorization. Here, 27 has 3³, while 225 only has 3², making divisibility impossible. Thus, 225 is not divisible by 27 because the power of 3 iRead more
The prime factorization of 225 is 3² x 5² and that of 27 is 3³. For divisibility, the prime factorization of the divisor must be included in the dividend’s factorization. Here, 27 has 3³, while 225 only has 3², making divisibility impossible. Thus, 225 is not divisible by 27 because the power of 3 in 225’s factorization is insufficient to accommodate the full factorization of 27.
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Is the first number divisible by the second? Use prime factorization. b) 96 and 24
The prime factorization of 96 is 2⁵ × 3, and that of 24 is 2³ ×3. Since all factors of 24 are present in 96’s factorization with equal or greater powers, 96 is divisible by 24. Dividing 96 by 24 confirms this: 96÷24 = 4 with no remainder. This demonstrates that the prime factors of 24 are fully inclRead more
The prime factorization of 96 is 2⁵ × 3, and that of 24 is 2³ ×3. Since all factors of 24 are present in 96’s factorization with equal or greater powers, 96 is divisible by 24. Dividing 96 by 24 confirms this: 96÷24 = 4 with no remainder. This demonstrates that the prime factors of 24 are fully included in those of 96, validating divisibility.
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Is the first number divisible by the second? Use prime factorization. c) 343 and 17
The prime factorization of 343 is 7³, indicating it is composed entirely of the prime number 7. Meanwhile, 17 is a prime number with no shared factors with 7. For divisibility, all prime factors of 17 must appear in 343’s factorization, which they do not. Hence, 343 is not divisible by 17, as theirRead more
The prime factorization of 343 is 7³, indicating it is composed entirely of the prime number 7. Meanwhile, 17 is a prime number with no shared factors with 7. For divisibility, all prime factors of 17 must appear in 343’s factorization, which they do not. Hence, 343 is not divisible by 17, as their factorization reveals no overlap or commonality in prime factors.
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Is the first number divisible by the second? Use prime factorization. d) 999 and 99
The prime factorization of 999 is 3³ x 37, and that of 99 is 3² × 11. For divisibility, the prime factors of 99 must appear in 999’s factorization. Here, 3² is present in 3³ and 11 divides evenly into 999. Performing the division confirms this: 999÷99=10. Thus, all factors of 99 are found in 999’s fRead more
The prime factorization of 999 is 3³ x 37, and that of 99 is 3² × 11. For divisibility, the prime factors of 99 must appear in 999’s factorization. Here, 3² is present in 3³ and 11 divides evenly into 999. Performing the division confirms this: 999÷99=10. Thus, all factors of 99 are found in 999’s factorization, verifying divisibility.
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Are 56 and 63 co-prime? Use prime factorization to verify.
To check co-primality, find the prime factorization of both numbers. For 56, it is 2³ × 7 for 63, it is 3² ×7. The shared factor 7 makes their greatest common divisor (GCD) 7, indicating they are not co-prime. Co-prime numbers must have no common factors other than 1, which is not the case here dueRead more
To check co-primality, find the prime factorization of both numbers. For 56, it is 2³ × 7 for 63, it is 3² ×7. The shared factor 7 makes their greatest common divisor (GCD) 7, indicating they are not co-prime. Co-prime numbers must have no common factors other than 1, which is not the case here due to their overlap in the factor 7.
For more NCERT Solutions for Class 6 Math Chapter 5 Prime Time Extra Questions and Answer:
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