Problem 7 Suppose b is any integer. Ifb mod 12 = 5, what is 6b mod 12? In other words, if division of b by 12 gives a remainder of 5, what is the remainder when 6b is divided by 12?

Q: Solve the following congruences: (a) 3x  8 (mod 5) (b) 3x  8 (mod 12)

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Q: Find a positive integer less than 8 for x that satisfies x = 720 + 3 (mod 7).

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Q: How many square roots of 3 (mod 1001) are there? (Hint: 1001 = 7 * 11 * 13)

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Q: Find all solutions of the given system of equations and check your answer graphically. (If there is…

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Q: Which irrational number can be multiplied by: 41 to get a product that equals 1 41 O -41

A: We can find the answer as below

Q: W found that 2,3,5,7 and 11 are not factors of 211. Can we conclude that 211 is prime? Why or why…

A: To check whether 211 is prime number on not.

Q: For a = 2695, b = 1820, use the prime factorizations to find lcm(a, b) and gcd(a, b). Verify that ab…

A: Given thata = 2695  and   b = 1820We have to find lcm(a, b) and gcd(a, b) by using prime…

Q: Simplify the following expressions by hand and show your work: 23^421 mod 31 57^2024! mod 43…

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Q: Give a non-zero numerical counterexample to the following statement: Every multiple of 12 is also a…

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Q: Find all solutions for x, up to congruence 7^78 (mod 79)

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Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: Prove: If the sum of the digits of a number in base bis divisible by b 1, so is the number.

A: Given statement: Sum of digits of any number (x) in any base b is divisible by b-1. To prove: The…

Q: Find 1112 mod 13.

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Q: Find the multiplicative inverse of a mod n, using 2 different methods. Show your steps in each case…

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Q: Find the square roots of 10 (mod 31).

A: As you asked more than one question I solved the first one.

Q: (a) 64x 56 (mod 52) (b) 14x = 35 (mod 56)

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Q: 0, the remainder is 2; and when divided by 11, the remainder is 0. remain is 5; when divided by

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Q: Add the bases and show work  2123 + 213 =

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Q: Does x2 + 5x congruent to 12 (mod 31) have a solution? (Hint: complete the square)

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Q: What is the inverse of 16 modulo 43 i.e. which number a has the property that 16*a has the remainder…

A: From question we have to find the inverse of 16 modulo 43. i.e  16*a mod(43)=1 It can also be…

Q: 14 x 75 10 4 70 x 10 = 70 x 4= 700 260 960 5 x 10 =D 5x4= 70 +. 50 20

A: in second column 70x4=280 not 260

Q: Find the solution of the following equations: 6x = 1(mod 33), 7x 1(mod 9)

A: If the given congruence equation is:.It is known that if a and m are relatively prime, then the…

Q: Find the remainder of 5130 modulo 28

A: Chinese Remainder Theorem : Let n1​,n2​,n3​,...,nk​∈N , with GCD(ni​,nj​)=1 for each i=j. Let…

Q: It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a…

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Q: b) Find an integer x such that 37x = 1 (mod 101).

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Q: Which of the following is an Integer ?

A: Given,           Ingeger number.