Consider $398,87x$ and $2$. Determine all possible digit replacements for $x$ so that the first number is divisible by the second.

Use the method of dividing by prime factors to find the least common multiple of $48$, $54$, and $60$.

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

$154,165$, and $2310$

Use the method of dividing by prime factors to find the least common multiple of $16$, $120$, and $216$.

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.

Answer each question concerning the Fibonacci sequence or the golden ratio.

From the pattern shown in the parity sequence, determine the parity of the one-hundredth term.

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.)

Determine whether the statement below is true or false.

There are infinitely many Fermat numbers.

Determine whether the statement below is true or false.

For every Mersenne prime, there is a corresponding perfect number.

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.

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 $11$.

$$29,630,419,088$$