Questions on Algebra: Divisibility and Prime Numbers answered by real tutors!

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Question 1209383: Prove that n^2-n is always even
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Question 1209383: Prove that n^2-n is always even
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Question 1209362: There are 830 composite numbers less than 1000. Let S be the set of composite numbers smaller than 1000 that are not divisible by 2, 3, or 7. How many elements does S have?
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Question 1209285: Find the LCM of 7, 10, 12, 15, 24, 75
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Question 1207615: As shown in class, the Euclidean algorithm can be used to find solutions to equations of the form
\[ax + by = c.\]
Use the Euclidean algorithm to find integers $x$ and $y$ such that $5x + 2y = 1,$ with the smallest possible positive value of $x$.
State your answer as a list with $x$ first and $y$ second, separated by a comma.
Note that while there are many pairs of integers $x$ and $y$ that satisfy this equation, there is only one pair that comes from using the Euclidean algorithm as described in class, and this pair solves the problem.
Click here to see answer by CPhill(1959)

Question 1208953: Suppose a >= 2 and n is a natural number larger than 1.
How can I prove that if n is odd, then a^n+1 is not prime?
Click here to see answer by ikleyn(52751)

Question 1208953: Suppose a >= 2 and n is a natural number larger than 1.
How can I prove that if n is odd, then a^n+1 is not prime?
Click here to see answer by math_tutor2020(3816)

Question 1208954: not sure how to prove this?

if n is odd then n^2 = 1 (mod 4)

thanks!
Click here to see answer by ikleyn(52751)

Question 1208954: not sure how to prove this?

if n is odd then n^2 = 1 (mod 4)

thanks!
Click here to see answer by math_tutor2020(3816)

Question 1208752: Show that x^2 + 7x + 13 is prime.
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Question 1208751: Show that x^2 + 3 is prime.
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Question 1208631: What is the greatest prime you must consider to test whether 4755 is prime?

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Question 1208348: Divide (x^5 - a^5) by (x - a)

Here is my set up:

(x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - a^5)/(x - a)

Is this correct?

Click here to see answer by ikleyn(52751)

Question 1207857: Find all integers , , such that is its own inverse modulo
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Question 1207737: Carlos wants to send Cecil a message encrypted with RSA. Cecil has published his public encryption exponent $4$ and his public modulus $15$. When Carlos encrypts the number $11$ with this system, what is the result?
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Question 1207738: The House of Lilliput is using RSA encryption to receive secret messages from all the realms. They have published their public encoding exponent $e = 2$ and their public modulus $M = pq = 15$.
Break the code: Find their secret decoding exponent $d.$
Click here to see answer by ikleyn(52751)

Question 1207739: Find an integer $x$ such that $0 \leq x < 205$ and $x^2 \equiv 11 \pmod{205}$.
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Question 1207739: Find an integer $x$ such that $0 \leq x < 205$ and $x^2 \equiv 11 \pmod{205}$.
Click here to see answer by greenestamps(13195)

Question 1207736: An eccentric baseball card collector wants to distribute her collection among her descendants. If she divided her cards among her 10 great-great-grandchildren, there would be 6 cards left over. If she divided her cards among her 7 great-grandchildren, there would be 5 cards left over. If she divided her cards among her 3 grandchildren, there would be 2 cards left over. If she divided her cards among her 2 children, there would be 0 cards left over.
What is the smallest possible number of cards in her collection?
Click here to see answer by greenestamps(13195)

Question 1207735: Greg finds the value of 708*709*710*711 and then divides the result over 712. What is the remainder? Is it possible to do this using mental math only?
Click here to see answer by math_tutor2020(3816)

Question 1207719: Let $m$ and $n$ be non-negative integers. If $m = 6n + 2$, then what integer between $0$ and $m$ is the inverse of $2$ modulo $m$? Answer in terms of $n$.
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Question 1207720: Find the number of solutions to
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{6}, \\
N &\equiv 2 \pmod{7}
in the interval 0 \le N < 1000.
Click here to see answer by greenestamps(13195)

Question 1207721: Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{7}.
Click here to see answer by greenestamps(13195)

Question 1207721: Find the smallest positive integer $N$ such that
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{7}.
Click here to see answer by ikleyn(52751)

Question 1207684: Find all integers $n$, $0 \le n < 163$, such that $n$ is its own inverse modulo $8.$
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Question 1207684: Find all integers $n$, $0 \le n < 163$, such that $n$ is its own inverse modulo $8.$
Click here to see answer by ikleyn(52751)

Question 1207685: The inverse of $a$ modulo $44$ is $b$. What is the inverse of $9$ modulo $10$?
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Question 1207686: Let $x$ and $y$ be integers. If $x$ and $y$ satisfy $41x + 5y = 31$, then find the residue of $x$ modulo $5$.
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Question 1207682: For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when divided by $131.$ What remainder does $n$ leave when divided by $131$?
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Question 1207683: Let $p$ be a prime. What are the possible remainders when $p$ is divided by $17?$ Select all that apply.
Click here to see answer by Edwin McCravy(20054)

Question 1207676: Find the remainder when 40^{13} is divided by 81.
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Question 1207676: Find the remainder when 40^{13} is divided by 81.
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Question 1207672: For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) + f(4).$
Click here to see answer by greenestamps(13195)

Question 1207670: Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.
Click here to see answer by math_tutor2020(3816)

Question 1207670: Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.
Click here to see answer by ikleyn(52751)

Question 1207671: Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.
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Question 1207654: A positive integer is called terrific if it has exactly $3$ positive divisors.
What is the smallest number of primes that could divide a terrific positive integer?
Click here to see answer by josgarithmetic(39613)

Question 1207656: A positive integer is called terrific if it has exactly $3$ positive divisors.
What is the smallest terrific positive integer?
Click here to see answer by greenestamps(13195)

Question 1207657: The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$
What is the smallest positive integer that has exactly $2$ perfect square divisors?
Click here to see answer by greenestamps(13195)

Question 1207613: For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
What is $\phi(5)$?
Click here to see answer by math_tutor2020(3816)

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295