Explain in your own words how to find the least common multiple of a group of numbers.

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

Explain in your own words how to find the greatest common factor of a group of numbers.

Give two factorizations of the number $75$ that are not prime factorizations.

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.

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 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.

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.

$200$ and $350$

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.

$945$ and $450$

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

$72$ and $90$

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.