Integers

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Problems about integers.

Use the definition of divides to decide whether the following statements are true or false. Explain your reasoning.

(a): $5|30$ TrueFalse
$30=5\cdot 6$ .
(b): $7|41$ TrueFalse
There is no integer solution to $41=7k$ . But $41=7\cdot 5+6$ .
(c): $0|3$ TrueFalse
There is no integer solution to $3=0k$ .
(d): $3|0$ TrueFalse
The solution to $0=3k$ is $k=0$ .
(e): $6|(2^2\cdot 3^4\cdot 5 \cdot 7)$ . TrueFalse
$6=2\cdot 3$ , and $2\cdot 3$ appears in factorization of the second number.
(f): $1000|(2^7\cdot 3^9\cdot 5^{11}\cdot 17^8)$ TrueFalse
$1000 = 2^3\cdot 5^3$ , and these primes appear enough times in factorization of the second number.
(g): $6000|(2^{21}\cdot 3^{17}\cdot 5^{89}\cdot 29^{20})$ . TrueFalse
$6000 = 2^4\cdot 3\cdot 5^3$ , and these primes appear enough times in factorization of the second number.

Factor the following integers. Enter the primes in increasing order, use * for multiplication, and do not use exponents. If the number is prime, enter the number itself. (Do not enter any spaces: Enter only digits and asterisks.)

(a): $15$ 3*5
(b): $12$ 2*2*3
(c): $111$ $\answer [format=string]{3*37}$
(d): $1234$ $\answer [format=string]{2*617}$
(e): $2345$ $\answer [format=string]{5*7*67}$
(f): $4567$ $\answer [format=string]{4567}$
(g): $111111$ $\answer [format=string]{3*7*11*13*37}$

If you use the math editor, you will need to delete spaces around each asterisk.

Find the greatest common divisors below:

(a): $\gcd (462,1463) = \answer {77}$
$462 = 2\cdot 3\cdot 7\cdot 11$ , and $1463 = 7\cdot 11\cdot 19$ .
(b): $\gcd (541,4669) = \answer {1}$ .
$541$ is prime. And $4669 = 7\cdot 23\cdot 29$ .
(c): $\gcd (10000,2^5\cdot 3^{19}\cdot 5^7\cdot 11^{13}) = \answer {10000}$
$10000 = 2^5\cdot 5^5$ .
(d): $\gcd (11111,2^{14}\cdot 7^{21}\cdot 41^{5}\cdot 101) = \answer {41}$
$11111 = 41\cdot 271$ .
(e): $\gcd (437^5,8993^3) = \answer {23^5}$
$437 = 19\cdot 23$ , and $8993=17\cdot 23^2$ .

Consider the following: $\begin{align*} 20 \div 8 &= 2 \text{ remainder }4, \\ 28 \div 12 &= 2 \text{ remainder }4. \end{align*}$

Is it correct to say that $20 \div 8 = 28 \div 12$ ? YesNo

Explain your reasoning.

The answer “ $2 \text { remainder }4$ ” is not a single number but rather a pair of numbers (a quotient and a remainder) that have different meanings. In particular, the 2 is about different things: groups of 8 versus groups of 12. Calling this pair of numbers “equal” is questionable.

Give a formula for the $n$ th even counting number: $\answer {2n}$

Give a formula for the $n$ th odd counting number: $\answer {2n-1}$ .

Give a formula for the $n$ th multiple of $3$ : $\answer {3n}$

Give a formula for the $n$ th multiple of $-7$ . $\answer {-7n}$

Give a formula for the $n$ th number whose remainder when divided by $5$ is $1$ .

If the first such number is 1, the formula is $\answer {5n-4}$ .

If the first such number is 6, the formula is $\answer {5n+1}$ .