The equation $17 + 51 = 68$ verifies Goldbach's conjecture for the number $68$. Decide whether the statement is true or false.

Disregarding A.M. or P.M., if it is now 3:00, what time of day will it be in 3500 hours?

Military Time Evaluate expression, where $+$ and $-$ indicate addition and subtraction, respectively, using military time. (Military time uses a 24-hour clock, where 2:00 A.M. is equivalent to 0200 hours and 10 P.M. is equivalent to 2200 hours.)

0600 $-$ 2200

The anterior pituitary gland does not produce __________.

A. Growth hormone (GH)

B. Adrenocorticotropic hormone (ACTH)

C. Oxytocin

D. Follicle-stimulating hormone (FSH)

Use rules of divisibility to determine whether each number given in is divisible by a. 2, b. 3, c. 4, d. 5, e. 6, f. 8, g. 9, h. 10, i. 12. 5.941.221

When applied to particular values of $n$, the factorial prime formula $n ! \pm 1$ produces no primes $\underline{\qquad\qquad}$ A. never. B. sometimes. C. always.

List two primes that are consecutive natural numbers. Can there be any others?

Rely on the previous examples to solve the following problem.

Jameel and Fahima work on an assembly line, inspecting electronic calculators. Jameel inspects the electronics of every sixteenth calculator, while Fahima inspects the workmanship of every thirty-sixth calculator. If they both start working at the same time, which calculator will be the first that they both inspect?

The following are the first nine Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, 34,.....a. Make a conjecture about each Fibonacci numbers after the first two. b. Write the next three numbers in the pattern. c. Research to find real-world example of this pattern.

The following array of numbers is called Pascal's triangle.

$$\begin{aligned} &1 \\ 1 ~~ & ~~~~ 1 \\ 1 ~\quad &2 \quad ~~1 \\ 1 ~~~~ 3 ~~ & ~~~~ 3 ~~~~~ 1 \\ 1 \quad 4~ \quad &6 \quad ~4 \quad ~1 \\ 1 ~~~~5~~~ 10~ & ~~~10 ~~~~5~~~~~ 1 \\ 1 \quad 6 \quad 15 \quad &20 \quad 15 \quad 6 \quad 1 \\ \end{aligned}$$

This array is important in the study of counting techniques and probability and appears in algebra in the binomial theorem. If the triangular array is written in a different form, as follows, and the sums along the diagonals as indicated by the dashed lines are found, there is an interesting occurrence. What do you find when the numbers are added?

$$\begin{array}{ccccccccc}1 & & & & & & \\ 1 & 1 & & & & & & \\ 1 & 2 & 1 & & & & \\ 1 & 3 & 3 & 1 & & & \\ 1 & 4 & 6 & 4 & 1 & & \\ 1 & 5 & 10 & 10 & 5 & 1 & \\ 1 & 6 & 15 & 20 & 15 & 6 & 1\end{array}$$

One of the authors has three sons who were born, from eldest to youngest, on August $30$, August $31$, and October $14$. For most (but not all) of each year, their ages are spaced two years from the eldest to the middle and three years from the middle to the youngest. In $2011$, their ages were three consecutive prime numbers for a period of exactly $44$ days. This same sittation had also occurred in exactly two previous years. Use this information to answer this exercise. (Hint: Consult the Sieve of Eratosthenes on the previous page for a listing of all primes less than $100$.)

Which were the "two previous years" referred to above?

Suppose that the least common multiple of $p$ and $q$ is $q$. What can we say about $p$ and $q$?

Choose any $6$-digit number consisting of three digits followed by the same three digits in the same order (for example, $467,467$). Divide by $13$. Divide by $11$. Divide by $7$. What do you notice? Why do you think this happens?

Explain why the factorial prime formula does not give twin primes for $n=2$.