a=0 (mod p) or b=0 (mod p).

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

A:

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

A:

Q: For which odd primes p is 5 a square mod p?

A:

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

A:

Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

A:

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

A:

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

A:

Q: If a = b (mod m) and a = b (mod n), and gcd (m,n) = 1, then a = b (mod mn) Select one: O True O…

A:

Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: () 2p n! (a) (") = 2 (mod p), where k for 0 <k <n. k!(n – k)! (b) (Зр - 1)! 2р2 (mod p3)

A: As per our guidelines we are supposed to answer only fast question. Kindly repost other part in the…

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

A:

Q: If a mod b = 0, then b | a true or false

A: It's  pronounced “a is congruent to b mod n” (or “a is congruent to b modulo n”). Basically, it…

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

A:

Q: Find a number a such that 70a = 1 mod 17.

A:

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: show that 2 is a primitive root mod 3^e for all e>=1

A:

Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

A:

Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

A:

Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

A:

Q: Show that y = x (mod p) leads to -y = -x (mod p).

A:

Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

A:

Q: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

A:

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A:

Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

A:

Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: Show that if a is an even integer, then a =0 (mod 4), and if a is an odd integer, then a =1 (mod 4).

A: Suppose a is an even integer Then a=2m for some integer m a2=(2m)2 a2=4m2Clearly 4|4so 4|4m2so…

Q: Prove or disprove: If a = b(mod n²) then a = b(mod n)

A:

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.

A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…

Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

A:

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.