Consider $985,23x$ and $6$. Determine all possible digit replacements for $x$ so that the first number is divisible by the second.

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 $90$

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.

$36$ and $90$

Consider the amicable pair $220$ and $284$. Determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair.

There are infinitely many prime numbers. Decide whether each statement is true or false.

Consider $64,537,84x$ and $4$. Determine all possible digit replacements for $x$ so that the first number is divisible by the second.

Is conjecture $(2)$ in the text (that the even perfect numbers end in the digits $6$ and $8$, alternately) true or false? Explain.

Consider $2,45x,765$ and $3$. Determine all possible digit replacements for $x$ so that the first number is divisible by the second.

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.

$77$ and $84$

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

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

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 $60$

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.

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