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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. \(k=32\)