If $p$ is a prime number, then the least common multiple of $p$ and $p^2$ is $p^3$. Decide whether the statement is true or false.

If $p$ and $q$ are different primes, $1$ is their greatest common factor and $pq$ is their least common multiple. Decide whether each statement is true or false.

The first four perfect numbers were identified in the text: $6$, $28$, $496$, and $8128$. The next two are $33,550,336$ and $8,589,869,056$. Verify that each of these six perfect numbers ends in either $6$ or $28$. (In fact, this is true of all even perfect numbers.)

Consider the divisibility test for the composite number $6$, and make a conjecture for the divisibility test for the composite number $15$.

There are only two solutions in integers for $a^2+4=b^3$. One solution is $a=2$, $b=2$. Find the other solution.

Use the divisibility test to determine whether each number is divisible by $11$.

$$29,630,419,088$$

There is only one solution in natural numbers for $a^2+2=b^3$, and it is $a=5$, $b=3$. Verify this solution.

Every odd prime can be expressed as the difference of two squares in one and only one way. (a) Find this one way for the prime number $5$. (b) Find this one way for the prime number $11$.

Use the divisibility test to determine whether each number is divisible by $11$.

$$60,128,459,358$$

Use the method of dividing by prime factors to determine the least common multiple of each group of numbers.

$63$ and $99$

Use this divisibility test to determine whether $108,410,313$ is divisible by $11$.

If $p$ is prime and the natural numbers $a$ and $p$ have no common factor except $1$, then $a^{p-1}-1$ is divisible by $p$. (a) Verify this for $p=5$ and $a=3$. (b) Verify this for $p=7$ and $a=2$.

Use this divisibility test to determine whether $6,524,846$ is divisible by $11$.

Use the method of dividing by prime factors to determine the least common multiple of each group of numbers.

$21$ and $56$