2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

A:

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Find a number a such that 0 ≤ a < 111 and (10270+ 1)35 = a (mod 111).

A:

Q: 1. Show that k1601 = k (mod 17) for any k.

A:

Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

A:

Q: 4. Consider the function f(x) = lIn(1 – x), defined for x E (-0, 1). (a) Find the Taylor polynomial…

A: Given :-  Consider the function f  x  = ln  1 - x  , defined for x ∈  - ∞ , 1 . (a) The Taylor…

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

A:

Q: 5. Using congruence theory (not brute force), find all solutions to the following linear congruence:…

A: Note:- A two variable linear congruence ax+by≡m    mod (n) has solution if and only if  gcd ( a, b,…

Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

A:

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

A:

Q: 1. Compute the least nonnegative residue of 10k + 1 modulo 13 for k 1, 2, 3, 4.

A:

Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

A:

Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: 14. Prove that if p and q are distinct primes, then p9-1+q-1=1 (mod pq).

A:

Q: How many positive integers n less than or equal to 11,347 satisfy d(n) 1 (mod 2)?

A: We need to find a number of positive integers n≤11347 such that d(n)≡1(mod 2) i.e., , k∈Z.So, we…

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

A:

Q: Let p and q be primes, Prove that plq if and only if p = = 9.

A:

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

A:

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

A:

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

A:

Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

A:

Q: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a &gt; 0.       We need to prove that a is even if and only if a ≡ 0…

Q: 6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the…

A:

Q: Let p be an odd prime and a, b are integers with pła, płb. Show that ab ()-(()) = Р

A:

Q: 1. Let m be a positive integer. If a = b (mod m) and c= d (mod m), a) Prove a+c= b + d (mod m) b)…

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: Determine the possibilities for the final digit of a sixth power of an integer.

A: The last digits of a number are 0,1,2,3,4,5,6,7,8 and 9. The number 0 raised to any number will be 0…