9. Evaluate the expressions: a. 41 div 5= b. 41 mod 5= 9. Last Tuesday you agreed to meet a friend exactly 120 days from that day. What day of the week will that day be? (Show your work) 10. Prove by contradiction: "For all integers m, 5m + 4 is not divisible by 5"

Q: Prove the following:For any natural number n, if n 2 + 6 n is an even number, then n is an even…

A: To prve that n is an even integer under the given conditions

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

A:

Q: which of the below integers is congruent to 5 modulo 10. O 15 O 20 O none l hu the hashing

A:

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: Determine whether the following statement is true or false. Justify your answer with a proof or a…

A:

Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

A:

Q: Which of these integers are in the congruence class of 11 mod 7? (Select all that apply.) O -11 4 6…

A: Which of these integers are in the congruence class of 11 mod 7? (Select all that apply.)

Q: (Adapted from Richardson problem 4.2.2, Solve the congruence 589.r = 494 (mod 817). Hint: The…

A:

Q: x is congruent to 43 mod 12. Then x is None of the option 91 75 89

A: Answer is 91.

Q: 2. Determine x >0 for each case: 5903 = x (mod 60), 17342 =r (mod 5).

A:

Q: which of the below integers is congruent to 6 modulo 12 O 40 O 34 O 30 O none

A:

Q: Consider 3 · 1, 3 · 2, 3 · 3, . . . , 3 · 12 all mod 13, or in other words, the multiples of 3 in…

A:

Q: 29. Decide whether each of these integers is congruent to 5 modulo 17. а) 80 с) —29 b) 103 d) –122

A:

Q: Is it possible, for some choice of positive whole numbers m and n, that 35m = 25? O The answer does…

A:

Q: 10. Find the value of 5368 (mod 41) Answer: Show your work here:

A: To find the value of- 5368(mod41)

Q: Determine which of the following statements are true: a. 54=72(mod16)

A:

Q: 16.5. Compute 2 (mod 9991) by successive squaring and use your answer to say whether you believe…

A:

Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: Prove the statement by the method of exhaustion: For each integer n with 1 less than or equal to n…

A:

Q: Consider the following statement. Is there an integer that has a remainder of 2 when it is divided…

A: The statement " Is there an integer that has a remainder of 2 when it is divided by 5 and a…

Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

A:

Q: 5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 ible by 7.

A:

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

A:

Q: Consider the following statement. For every integer m, 7m + 4 is not divisible by 7. Construct a…

A: Given statement is for every integer m,7m+4 is not divisible by 7. We prove this by contradiction.

Q: Prove the following (by using contrapositive): Let a, b and n be integers. If n doesn’t divide ab,…

A:

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: II. Determine whether each of these integers are congruent to 7 mod 9 4. 80 5. 106 6. -29

A: we know that x=y mod(n)  where Y is remainder

Q: d) If 72 = 1 (mod 5), with 0<r <4, then z =. e) The number of (positive) divisors of 23345 is

A:

Q: which of the below integers is congruent to 6 modulo 12 O none 40 30 34

A: 30

Q: Prove directly from the definitions that for all integers m and n, if m and n have the same parity,…

A:

Q: 4. Let a, b, and d be integers such that dļab. Is it true that either da or d b? If you claim that…

A: Given integers a,b and d. We need to check whether if d/

Q: State as TRUE or FALSE. d) 143 = -40 (mod 5) e) –4 = 788 (mod 3)

A: a≡b(mod n) means a-b is divisible by n

Q: Suppose that a = 3 mod 12 and b =7 mod 12 IfC = 3 a – 2 b mod 12 then C =

A:

Q: The value of 52003 mod 7 is? (3 4 8

A:

Q: 5. The value of 693 mod 11 is O 3 O 7 O 6 4

A:

Q: Evaluate the expressions. (a) 48 div 5 (b) 48 mod 5

A: (a) The given expression 48 div 5. Explanation: Consider the division. 5948453 That is, 48 = 9(5)…

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: What is the result of: 6(40 – 13) (mod 11) The result is 8

A: Given 6(40-13)(mod 11)

Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: Find an integer x that satisfies the equation. 5* = 4(mod 11) 17x?=10(mod 29) i. %3D ii.

A:

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: 5. Decide which of these integers is congruent to 13 modulo 17. a) 81 b) 18 c) -38 d) 83 e) -11

A:

Q: 3. If x mod 4 = 1 and y mod 4 = 3, then xy could be O 901 O 299 O 508 O 113

A: We shall use the division algorithm

Q: 4. Solve the following: 34x 60 (mod98) 5. Detemine all the possible positive m values greater than 1…

A:

Q: x is congruent to 21 mod 8. Then x is None of the option 29 16 24

A:

Q: Suppose a, b, c and d are integers. Prove that  if a divides b and c divides d, then ac divides bd

A: According to the given information:   It is given that a divides b, there is an integer x such that…

Q: Prove that if an integer m has the same parity as an integer n, 5m + 7n is even.

A:

Q: Calculate (13, 000 +2)10 (mod 13). Find the omeir lor of 3456 whon dlivided by the following Hint:…

A:

Q: 19. Which of the following pairs of integers is NOT congruent to modulo 9? 1 and 10 4 and -32 -2 and…

A: Given  19) Congruent to modulo 9 20) 24 is congruent to 58 mod n 21) 1220(mod 25)

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.