6.E: Prime Numbers (Exercises)
- Page ID
- 7605
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Exercise \(\PageIndex{1}\):
Are 253 and 257 prime?
Exercise \(\PageIndex{2}\):
Using prime factorization find the GCD and LCM of 3920 and 820.
Exercise \(\PageIndex{3}\):
Using prime factorization find the GCD and LCM of \(30030\) and \(165\).
Exercise \(\PageIndex{4}\):
Find the prime factorization of \(10101\).
- Answer
-
\(3 \times 7 \times 13 \times 37 \).
Exercise \(\PageIndex{5}\):
1. Let \(a\) and \(b\) be positive integers such that \(a^2|b^2\). Show that \(a|b\).
2. Let \(a\) and \(b\) be positive integers. Prove that \(\gcd (a^2,b^2)=(\gcd(a,b)^2\).
3. Let \(a\) and \(b\) be positive integers such that \(\gcd(a,b)=1\). If \(ab\) is a perfect square then show that \(a\) and \(b\) are both perfect square.
- Hint
-
Use the fundamental theorem of Arithmetic.
Exercise \(\PageIndex{6}\):
Describe in terms of the prime numbers all numbers with exactly four divisors.
Exercise \(\PageIndex{7}\):
1. Find a prime \(k\) such that \( 2^k-1\) is not a prime.
2. Find an integer \(k\), which is a power of 2 such that \( 2^k+1\) is not a prime.
- Answer
-
1. k=29, 2. \(k=32\)