6.E: Prime Numbers (Exercises)

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. $k=32$