Use the formula given on the previous page and the results of the previous exercises to find the least common multiple of each group of numbers.

$18$ and $60$

Is conjecture $(1)$ in the text (that the $n$th perfect number contains exactly $n$ digits) true or false? Explain.

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.

Determine whether the statement below is true or false.

There are infinitely many Fermat numbers.

Summarize the effects of imperialism on nineteenth-century China.

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.

There are no even prime numbers. Decide whether the statement is true or false.

Determine whether the statement below is true or false.

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

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.)

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

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

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

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

$$29,630,419,088$$